Which Of The Following Represents The Solution To The Inequality 2 ∣ 5 − 2 X ∣ − 3 ≤ 15 2|5-2x|-3 \leq 15 2∣5 − 2 X ∣ − 3 ≤ 15 ?A. ( − ∞ , − 2 ) ∪ ( 7 , ∞ (-\infty,-2) \cup (7, \infty ( − ∞ , − 2 ) ∪ ( 7 , ∞ ]B. ( − ∞ , 15 ) ∪ ( 7.5 , ∞ (-\infty, 15) \cup (7.5, \infty ( − ∞ , 15 ) ∪ ( 7.5 , ∞ ]C. 1 − 2.7 1-2.7 1 − 2.7 D. 1.5 , 7.5 {1.5, 7.5} 1.5 , 7.5

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Introduction

Absolute value inequalities can be challenging to solve, but with a clear understanding of the concept and a step-by-step approach, they can be tackled with ease. In this article, we will explore the solution to the inequality 2∣5−2x∣−3≤152|5-2x|-3 \leq 15 and provide a detailed explanation of the process involved.

Understanding Absolute Value Inequalities

Absolute value inequalities involve the absolute value of an expression, which is always non-negative. The absolute value of a number aa is denoted by ∣a∣|a| and is defined as:

∣a∣={aif a≥0−aif a<0|a| = \begin{cases} a & \text{if } a \geq 0 \\ -a & \text{if } a < 0 \end{cases}

When solving absolute value inequalities, we need to consider two cases: one where the expression inside the absolute value is non-negative and another where it is negative.

Solving the Inequality

To solve the inequality 2∣5−2x∣−3≤152|5-2x|-3 \leq 15, we will follow the steps below:

Step 1: Isolate the Absolute Value Expression

The first step is to isolate the absolute value expression on one side of the inequality. We can do this by adding 3 to both sides of the inequality:

2∣5−2x∣≤182|5-2x| \leq 18

Step 2: Divide by 2

Next, we divide both sides of the inequality by 2 to isolate the absolute value expression:

∣5−2x∣≤9|5-2x| \leq 9

Step 3: Split into Two Cases

Now, we split the inequality into two cases: one where the expression inside the absolute value is non-negative and another where it is negative.

Case 1: 5−2x≥05-2x \geq 0

In this case, the absolute value expression can be rewritten as:

5−2x≤95-2x \leq 9

Subtracting 5 from both sides gives:

−2x≤4-2x \leq 4

Dividing both sides by -2 (and reversing the inequality sign) gives:

x≥−2x \geq -2

However, since we are considering the case where 5−2x≥05-2x \geq 0, we need to find the values of xx that satisfy this condition. Solving the inequality 5−2x≥05-2x \geq 0 gives:

x≤52x \leq \frac{5}{2}

Therefore, the solution to this case is:

−2≤x≤52-2 \leq x \leq \frac{5}{2}

Case 2: 5−2x<05-2x < 0

In this case, the absolute value expression can be rewritten as:

−(5−2x)≤9-(5-2x) \leq 9

Simplifying the expression gives:

−5+2x≤9-5+2x \leq 9

Adding 5 to both sides gives:

2x≤142x \leq 14

Dividing both sides by 2 gives:

x≤7x \leq 7

However, since we are considering the case where 5−2x<05-2x < 0, we need to find the values of xx that satisfy this condition. Solving the inequality 5−2x<05-2x < 0 gives:

x>52x > \frac{5}{2}

Therefore, the solution to this case is:

52<x≤7\frac{5}{2} < x \leq 7

Combining the Solutions

Now, we need to combine the solutions from both cases. The solution to the inequality 2∣5−2x∣−3≤152|5-2x|-3 \leq 15 is the union of the solutions from both cases:

−2≤x≤52∪52<x≤7-2 \leq x \leq \frac{5}{2} \cup \frac{5}{2} < x \leq 7

Simplifying the expression gives:

−2≤x≤7-2 \leq x \leq 7

However, we need to consider the values of xx that satisfy the condition 5−2x≥05-2x \geq 0. Solving this inequality gives:

x≤52x \leq \frac{5}{2}

Therefore, the solution to the inequality 2∣5−2x∣−3≤152|5-2x|-3 \leq 15 is:

−2≤x≤52-2 \leq x \leq \frac{5}{2}

Conclusion

In this article, we have solved the absolute value inequality 2∣5−2x∣−3≤152|5-2x|-3 \leq 15 using a step-by-step approach. We have considered two cases: one where the expression inside the absolute value is non-negative and another where it is negative. By combining the solutions from both cases, we have found the solution to the inequality to be −2≤x≤52-2 \leq x \leq \frac{5}{2}.

Final Answer

The final answer is:

\boxed{(-\infty,-2) \cup (7, \infty)}$<br/> # Frequently Asked Questions: Solving Absolute Value Inequalities =====================================================

Q: What is an absolute value inequality?

A: An absolute value inequality is an inequality that involves the absolute value of an expression. The absolute value of a number aa is denoted by ∣a∣|a| and is defined as:

∣a∣={aamp;if a≥0−aamp;if alt;0</span></p><h2>Q:HowdoIsolveanabsolutevalueinequality?</h2><hr><p>A:Tosolveanabsolutevalueinequality,youneedtoconsidertwocases:onewheretheexpressioninsidetheabsolutevalueisnon−negativeandanotherwhereitisnegative.Youcanthenusethepropertiesofabsolutevaluetosimplifytheinequalityandfindthesolution.</p><h2>Q:Whatarethepropertiesofabsolutevalue?</h2><hr><p>A:Thepropertiesofabsolutevalueare:</p><ul><li><spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mimathvariant="normal">∣</mi><mi>a</mi><mimathvariant="normal">∣</mi><mo>≥</mo><mn>0</mn></mrow><annotationencoding="application/x−tex">∣a∣≥0</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:1em;vertical−align:−0.25em;"></span><spanclass="mord">∣</span><spanclass="mordmathnormal">a</span><spanclass="mord">∣</span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mrel">≥</span><spanclass="mspace"style="margin−right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:0.6444em;"></span><spanclass="mord">0</span></span></span></span>forallrealnumbers<spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi></mrow><annotationencoding="application/x−tex">a</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:0.4306em;"></span><spanclass="mordmathnormal">a</span></span></span></span></li><li><spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mimathvariant="normal">∣</mi><mi>a</mi><mi>b</mi><mimathvariant="normal">∣</mi><mo>=</mo><mimathvariant="normal">∣</mi><mi>a</mi><mimathvariant="normal">∣</mi><mimathvariant="normal">∣</mi><mi>b</mi><mimathvariant="normal">∣</mi></mrow><annotationencoding="application/x−tex">∣ab∣=∣a∣∣b∣</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:1em;vertical−align:−0.25em;"></span><spanclass="mord">∣</span><spanclass="mordmathnormal">ab</span><spanclass="mord">∣</span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="margin−right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:1em;vertical−align:−0.25em;"></span><spanclass="mord">∣</span><spanclass="mordmathnormal">a</span><spanclass="mord">∣∣</span><spanclass="mordmathnormal">b</span><spanclass="mord">∣</span></span></span></span>forallrealnumbers<spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi></mrow><annotationencoding="application/x−tex">a</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:0.4306em;"></span><spanclass="mordmathnormal">a</span></span></span></span>and<spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>b</mi></mrow><annotationencoding="application/x−tex">b</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:0.6944em;"></span><spanclass="mordmathnormal">b</span></span></span></span></li><li><spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mimathvariant="normal">∣</mi><mi>a</mi><mo>+</mo><mi>b</mi><mimathvariant="normal">∣</mi><mo>≤</mo><mimathvariant="normal">∣</mi><mi>a</mi><mimathvariant="normal">∣</mi><mo>+</mo><mimathvariant="normal">∣</mi><mi>b</mi><mimathvariant="normal">∣</mi></mrow><annotationencoding="application/x−tex">∣a+b∣≤∣a∣+∣b∣</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:1em;vertical−align:−0.25em;"></span><spanclass="mord">∣</span><spanclass="mordmathnormal">a</span><spanclass="mspace"style="margin−right:0.2222em;"></span><spanclass="mbin">+</span><spanclass="mspace"style="margin−right:0.2222em;"></span></span><spanclass="base"><spanclass="strut"style="height:1em;vertical−align:−0.25em;"></span><spanclass="mordmathnormal">b</span><spanclass="mord">∣</span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mrel">≤</span><spanclass="mspace"style="margin−right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:1em;vertical−align:−0.25em;"></span><spanclass="mord">∣</span><spanclass="mordmathnormal">a</span><spanclass="mord">∣</span><spanclass="mspace"style="margin−right:0.2222em;"></span><spanclass="mbin">+</span><spanclass="mspace"style="margin−right:0.2222em;"></span></span><spanclass="base"><spanclass="strut"style="height:1em;vertical−align:−0.25em;"></span><spanclass="mord">∣</span><spanclass="mordmathnormal">b</span><spanclass="mord">∣</span></span></span></span>forallrealnumbers<spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi></mrow><annotationencoding="application/x−tex">a</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:0.4306em;"></span><spanclass="mordmathnormal">a</span></span></span></span>and<spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>b</mi></mrow><annotationencoding="application/x−tex">b</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:0.6944em;"></span><spanclass="mordmathnormal">b</span></span></span></span></li></ul><h2>Q:HowdoIisolatetheabsolutevalueexpressioninaninequality?</h2><hr><p>A:Toisolatetheabsolutevalueexpressioninaninequality,youcanaddorsubtractthesamevaluefrombothsidesoftheinequality.Forexample,ifyouhavetheinequality<spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mimathvariant="normal">∣</mi><mi>x</mi><mimathvariant="normal">∣</mi><mo>+</mo><mn>2</mn><mo>≤</mo><mn>5</mn></mrow><annotationencoding="application/x−tex">∣x∣+2≤5</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:1em;vertical−align:−0.25em;"></span><spanclass="mord">∣</span><spanclass="mordmathnormal">x</span><spanclass="mord">∣</span><spanclass="mspace"style="margin−right:0.2222em;"></span><spanclass="mbin">+</span><spanclass="mspace"style="margin−right:0.2222em;"></span></span><spanclass="base"><spanclass="strut"style="height:0.7804em;vertical−align:−0.136em;"></span><spanclass="mord">2</span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mrel">≤</span><spanclass="mspace"style="margin−right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:0.6444em;"></span><spanclass="mord">5</span></span></span></span>,youcansubtract2frombothsidestoget<spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mimathvariant="normal">∣</mi><mi>x</mi><mimathvariant="normal">∣</mi><mo>≤</mo><mn>3</mn></mrow><annotationencoding="application/x−tex">∣x∣≤3</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:1em;vertical−align:−0.25em;"></span><spanclass="mord">∣</span><spanclass="mordmathnormal">x</span><spanclass="mord">∣</span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mrel">≤</span><spanclass="mspace"style="margin−right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:0.6444em;"></span><spanclass="mord">3</span></span></span></span>.</p><h2>Q:Whatisthedifferencebetweenanabsolutevalueinequalityandalinearinequality?</h2><hr><p>A:Anabsolutevalueinequalityisaninequalitythatinvolvestheabsolutevalueofanexpression,whilealinearinequalityisaninequalitythatinvolvesalinearexpression.Forexample,theinequality<spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mimathvariant="normal">∣</mi><mi>x</mi><mimathvariant="normal">∣</mi><mo>≤</mo><mn>3</mn></mrow><annotationencoding="application/x−tex">∣x∣≤3</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:1em;vertical−align:−0.25em;"></span><spanclass="mord">∣</span><spanclass="mordmathnormal">x</span><spanclass="mord">∣</span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mrel">≤</span><spanclass="mspace"style="margin−right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:0.6444em;"></span><spanclass="mord">3</span></span></span></span>isanabsolutevalueinequality,whiletheinequality<spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi><mo>≤</mo><mn>3</mn></mrow><annotationencoding="application/x−tex">x≤3</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:0.7719em;vertical−align:−0.136em;"></span><spanclass="mordmathnormal">x</span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mrel">≤</span><spanclass="mspace"style="margin−right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:0.6444em;"></span><spanclass="mord">3</span></span></span></span>isalinearinequality.</p><h2>Q:CanIusethesamemethodstosolveabsolutevalueinequalitiesasIwouldtosolvelinearinequalities?</h2><hr><p>A:No,youcannotusethesamemethodstosolveabsolutevalueinequalitiesasyouwouldtosolvelinearinequalities.Absolutevalueinequalitiesrequireadifferentapproach,astheyinvolvetheabsolutevalueofanexpression.</p><h2>Q:Whataresomecommonmistakestoavoidwhensolvingabsolutevalueinequalities?</h2><hr><p>A:Somecommonmistakestoavoidwhensolvingabsolutevalueinequalitiesinclude:</p><ul><li>Failingtoconsiderbothcases(whentheexpressioninsidetheabsolutevalueisnon−negativeandwhenitisnegative)</li><li>Notisolatingtheabsolutevalueexpressioncorrectly</li><li>Notusingthepropertiesofabsolutevaluecorrectly</li><li>Notcheckingthesolutiontomakesureitsatisfiestheoriginalinequality</li></ul><h2>Q:HowcanIcheckmysolutiontoanabsolutevalueinequality?</h2><hr><p>A:Tocheckyoursolutiontoanabsolutevalueinequality,youcanplugthesolutionbackintotheoriginalinequalityandverifythatitistrue.Youcanalsouseagraphingcalculatororacomputeralgebrasystemtocheckyoursolution.</p><h2>Q:Whataresomereal−worldapplicationsofabsolutevalueinequalities?</h2><hr><p>A:Absolutevalueinequalitieshavemanyreal−worldapplications,including:</p><ul><li>Modelingthedistancebetweentwopoints</li><li>Findingthemaximumorminimumvalueofafunction</li><li>Solvingproblemsinvolvingmotionorvelocity</li><li>Modelingthecostorrevenueofabusiness</li></ul><h2>Q:CanIuseabsolutevalueinequalitiestosolveproblemsinvolvingfinanceoreconomics?</h2><hr><p>A:Yes,youcanuseabsolutevalueinequalitiestosolveproblemsinvolvingfinanceoreconomics.Forexample,youcanuseabsolutevalueinequalitiestomodelthecostorrevenueofabusiness,ortofindthemaximumorminimumvalueofafunctionthatrepresentsafinancialoreconomicquantity.</p><h2>Q:Whataresomecommontypesofabsolutevalueinequalities?</h2><hr><p>A:Somecommontypesofabsolutevalueinequalitiesinclude:</p><ul><li><spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mimathvariant="normal">∣</mi><mi>x</mi><mimathvariant="normal">∣</mi><mo>≤</mo><mi>a</mi></mrow><annotationencoding="application/x−tex">∣x∣≤a</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:1em;vertical−align:−0.25em;"></span><spanclass="mord">∣</span><spanclass="mordmathnormal">x</span><spanclass="mord">∣</span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mrel">≤</span><spanclass="mspace"style="margin−right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:0.4306em;"></span><spanclass="mordmathnormal">a</span></span></span></span></li><li><spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mimathvariant="normal">∣</mi><mi>x</mi><mimathvariant="normal">∣</mi><mo>≥</mo><mi>a</mi></mrow><annotationencoding="application/x−tex">∣x∣≥a</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:1em;vertical−align:−0.25em;"></span><spanclass="mord">∣</span><spanclass="mordmathnormal">x</span><spanclass="mord">∣</span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mrel">≥</span><spanclass="mspace"style="margin−right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:0.4306em;"></span><spanclass="mordmathnormal">a</span></span></span></span></li><li><spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mimathvariant="normal">∣</mi><mi>x</mi><mo>−</mo><mi>a</mi><mimathvariant="normal">∣</mi><mo>≤</mo><mi>b</mi></mrow><annotationencoding="application/x−tex">∣x−a∣≤b</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:1em;vertical−align:−0.25em;"></span><spanclass="mord">∣</span><spanclass="mordmathnormal">x</span><spanclass="mspace"style="margin−right:0.2222em;"></span><spanclass="mbin">−</span><spanclass="mspace"style="margin−right:0.2222em;"></span></span><spanclass="base"><spanclass="strut"style="height:1em;vertical−align:−0.25em;"></span><spanclass="mordmathnormal">a</span><spanclass="mord">∣</span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mrel">≤</span><spanclass="mspace"style="margin−right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:0.6944em;"></span><spanclass="mordmathnormal">b</span></span></span></span></li><li><spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mimathvariant="normal">∣</mi><mi>x</mi><mo>−</mo><mi>a</mi><mimathvariant="normal">∣</mi><mo>≥</mo><mi>b</mi></mrow><annotationencoding="application/x−tex">∣x−a∣≥b</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:1em;vertical−align:−0.25em;"></span><spanclass="mord">∣</span><spanclass="mordmathnormal">x</span><spanclass="mspace"style="margin−right:0.2222em;"></span><spanclass="mbin">−</span><spanclass="mspace"style="margin−right:0.2222em;"></span></span><spanclass="base"><spanclass="strut"style="height:1em;vertical−align:−0.25em;"></span><spanclass="mordmathnormal">a</span><spanclass="mord">∣</span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mrel">≥</span><spanclass="mspace"style="margin−right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:0.6944em;"></span><spanclass="mordmathnormal">b</span></span></span></span></li></ul><h2>Q:HowcanIuseabsolutevalueinequalitiestosolveproblemsinvolvingscienceorengineering?</h2><hr><p>A:Youcanuseabsolutevalueinequalitiestosolveproblemsinvolvingscienceorengineeringbymodelingthedistancebetweentwopoints,findingthemaximumorminimumvalueofafunction,orsolvingproblemsinvolvingmotionorvelocity.</p><h2>Q:Whataresometipsforsolvingabsolutevalueinequalities?</h2><hr><p>A:Sometipsforsolvingabsolutevalueinequalitiesinclude:</p><ul><li>Readtheproblemcarefullyandunderstandwhatisbeingasked</li><li>Usethepropertiesofabsolutevaluecorrectly</li><li>Considerbothcases(whentheexpressioninsidetheabsolutevalueisnon−negativeandwhenitisnegative)</li><li>Checkyoursolutiontomakesureitsatisfiestheoriginalinequality</li><li>Useagraphingcalculatororacomputeralgebrasystemtocheckyoursolution.</li></ul>|a| = \begin{cases} a &amp; \text{if } a \geq 0 \\ -a &amp; \text{if } a &lt; 0 \end{cases} </span></p> <h2>Q: How do I solve an absolute value inequality?</h2> <hr> <p>A: To solve an absolute value inequality, you need to consider two cases: one where the expression inside the absolute value is non-negative and another where it is negative. You can then use the properties of absolute value to simplify the inequality and find the solution.</p> <h2>Q: What are the properties of absolute value?</h2> <hr> <p>A: The properties of absolute value are:</p> <ul> <li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∣</mi><mi>a</mi><mi mathvariant="normal">∣</mi><mo>≥</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">|a| \geq 0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">∣</span><span class="mord mathnormal">a</span><span class="mord">∣</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≥</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span> for all real numbers <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi></mrow><annotation encoding="application/x-tex">a</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">a</span></span></span></span></li> <li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∣</mi><mi>a</mi><mi>b</mi><mi mathvariant="normal">∣</mi><mo>=</mo><mi mathvariant="normal">∣</mi><mi>a</mi><mi mathvariant="normal">∣</mi><mi mathvariant="normal">∣</mi><mi>b</mi><mi mathvariant="normal">∣</mi></mrow><annotation encoding="application/x-tex">|ab| = |a||b|</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">∣</span><span class="mord mathnormal">ab</span><span class="mord">∣</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">∣</span><span class="mord mathnormal">a</span><span class="mord">∣∣</span><span class="mord mathnormal">b</span><span class="mord">∣</span></span></span></span> for all real numbers <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi></mrow><annotation encoding="application/x-tex">a</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">a</span></span></span></span> and <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>b</mi></mrow><annotation encoding="application/x-tex">b</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">b</span></span></span></span></li> <li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∣</mi><mi>a</mi><mo>+</mo><mi>b</mi><mi mathvariant="normal">∣</mi><mo>≤</mo><mi mathvariant="normal">∣</mi><mi>a</mi><mi mathvariant="normal">∣</mi><mo>+</mo><mi mathvariant="normal">∣</mi><mi>b</mi><mi mathvariant="normal">∣</mi></mrow><annotation encoding="application/x-tex">|a+b| \leq |a| + |b|</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">∣</span><span class="mord mathnormal">a</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">b</span><span class="mord">∣</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">∣</span><span class="mord mathnormal">a</span><span class="mord">∣</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">∣</span><span class="mord mathnormal">b</span><span class="mord">∣</span></span></span></span> for all real numbers <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi></mrow><annotation encoding="application/x-tex">a</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">a</span></span></span></span> and <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>b</mi></mrow><annotation encoding="application/x-tex">b</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">b</span></span></span></span></li> </ul> <h2>Q: How do I isolate the absolute value expression in an inequality?</h2> <hr> <p>A: To isolate the absolute value expression in an inequality, you can add or subtract the same value from both sides of the inequality. For example, if you have the inequality <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∣</mi><mi>x</mi><mi mathvariant="normal">∣</mi><mo>+</mo><mn>2</mn><mo>≤</mo><mn>5</mn></mrow><annotation encoding="application/x-tex">|x| + 2 \leq 5</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">∣</span><span class="mord mathnormal">x</span><span class="mord">∣</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.7804em;vertical-align:-0.136em;"></span><span class="mord">2</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">5</span></span></span></span>, you can subtract 2 from both sides to get <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∣</mi><mi>x</mi><mi mathvariant="normal">∣</mi><mo>≤</mo><mn>3</mn></mrow><annotation encoding="application/x-tex">|x| \leq 3</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">∣</span><span class="mord mathnormal">x</span><span class="mord">∣</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">3</span></span></span></span>.</p> <h2>Q: What is the difference between an absolute value inequality and a linear inequality?</h2> <hr> <p>A: An absolute value inequality is an inequality that involves the absolute value of an expression, while a linear inequality is an inequality that involves a linear expression. For example, the inequality <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∣</mi><mi>x</mi><mi mathvariant="normal">∣</mi><mo>≤</mo><mn>3</mn></mrow><annotation encoding="application/x-tex">|x| \leq 3</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">∣</span><span class="mord mathnormal">x</span><span class="mord">∣</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">3</span></span></span></span> is an absolute value inequality, while the inequality <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi><mo>≤</mo><mn>3</mn></mrow><annotation encoding="application/x-tex">x \leq 3</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7719em;vertical-align:-0.136em;"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">3</span></span></span></span> is a linear inequality.</p> <h2>Q: Can I use the same methods to solve absolute value inequalities as I would to solve linear inequalities?</h2> <hr> <p>A: No, you cannot use the same methods to solve absolute value inequalities as you would to solve linear inequalities. Absolute value inequalities require a different approach, as they involve the absolute value of an expression.</p> <h2>Q: What are some common mistakes to avoid when solving absolute value inequalities?</h2> <hr> <p>A: Some common mistakes to avoid when solving absolute value inequalities include:</p> <ul> <li>Failing to consider both cases (when the expression inside the absolute value is non-negative and when it is negative)</li> <li>Not isolating the absolute value expression correctly</li> <li>Not using the properties of absolute value correctly</li> <li>Not checking the solution to make sure it satisfies the original inequality</li> </ul> <h2>Q: How can I check my solution to an absolute value inequality?</h2> <hr> <p>A: To check your solution to an absolute value inequality, you can plug the solution back into the original inequality and verify that it is true. You can also use a graphing calculator or a computer algebra system to check your solution.</p> <h2>Q: What are some real-world applications of absolute value inequalities?</h2> <hr> <p>A: Absolute value inequalities have many real-world applications, including:</p> <ul> <li>Modeling the distance between two points</li> <li>Finding the maximum or minimum value of a function</li> <li>Solving problems involving motion or velocity</li> <li>Modeling the cost or revenue of a business</li> </ul> <h2>Q: Can I use absolute value inequalities to solve problems involving finance or economics?</h2> <hr> <p>A: Yes, you can use absolute value inequalities to solve problems involving finance or economics. For example, you can use absolute value inequalities to model the cost or revenue of a business, or to find the maximum or minimum value of a function that represents a financial or economic quantity.</p> <h2>Q: What are some common types of absolute value inequalities?</h2> <hr> <p>A: Some common types of absolute value inequalities include:</p> <ul> <li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∣</mi><mi>x</mi><mi mathvariant="normal">∣</mi><mo>≤</mo><mi>a</mi></mrow><annotation encoding="application/x-tex">|x| \leq a</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">∣</span><span class="mord mathnormal">x</span><span class="mord">∣</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">a</span></span></span></span></li> <li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∣</mi><mi>x</mi><mi mathvariant="normal">∣</mi><mo>≥</mo><mi>a</mi></mrow><annotation encoding="application/x-tex">|x| \geq a</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">∣</span><span class="mord mathnormal">x</span><span class="mord">∣</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≥</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">a</span></span></span></span></li> <li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∣</mi><mi>x</mi><mo>−</mo><mi>a</mi><mi mathvariant="normal">∣</mi><mo>≤</mo><mi>b</mi></mrow><annotation encoding="application/x-tex">|x-a| \leq b</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">∣</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">a</span><span class="mord">∣</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">b</span></span></span></span></li> <li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∣</mi><mi>x</mi><mo>−</mo><mi>a</mi><mi mathvariant="normal">∣</mi><mo>≥</mo><mi>b</mi></mrow><annotation encoding="application/x-tex">|x-a| \geq b</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">∣</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">a</span><span class="mord">∣</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≥</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">b</span></span></span></span></li> </ul> <h2>Q: How can I use absolute value inequalities to solve problems involving science or engineering?</h2> <hr> <p>A: You can use absolute value inequalities to solve problems involving science or engineering by modeling the distance between two points, finding the maximum or minimum value of a function, or solving problems involving motion or velocity.</p> <h2>Q: What are some tips for solving absolute value inequalities?</h2> <hr> <p>A: Some tips for solving absolute value inequalities include:</p> <ul> <li>Read the problem carefully and understand what is being asked</li> <li>Use the properties of absolute value correctly</li> <li>Consider both cases (when the expression inside the absolute value is non-negative and when it is negative)</li> <li>Check your solution to make sure it satisfies the original inequality</li> <li>Use a graphing calculator or a computer algebra system to check your solution.</li> </ul>