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. ( − ∞ , 1.5 ) ∪ ( 7.5 , ∞ (-\infty, 1.5) \cup (7.5, \infty ( − ∞ , 1.5 ) ∪ ( 7.5 , ∞ ] C. − 2 , 7 {-2, 7} − 2 , 7 D. 1.5 , 7.5 {1.5, 7.5} 1.5 , 7.5

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Introduction

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Inequalities with absolute values 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 focus on solving the inequality $2|5-2x|-3 \leq 15$ and explore the different solution sets that can be obtained.

Understanding Absolute Values

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Absolute values represent the distance of a number from zero on the number line. For any real number $a$, the absolute value of $a$ is denoted by $|a|$ and is defined as:

$$|a| = \begin{cases} a & \text{if } a \geq 0 \\ -a & \text{if } a < 0 \end{cases}$$

Solving the Inequality

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To solve the inequality $2|5-2x|-3 \leq 15$, we need to isolate the absolute value expression. We can start by adding 3 to both sides of the inequality:

$$2|5-2x| \leq 18$$

Next, we can divide both sides of the inequality by 2:

$$|5-2x| \leq 9$$

Case 1: $5-2x \geq 0$

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When $5-2x \geq 0$, the absolute value expression can be rewritten as:

$$5-2x \leq 9$$

Subtracting 5 from both sides of the inequality gives us:

$$-2x \leq 4$$

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

$$x \geq -2$$

Case 2: $5-2x < 0$

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When $5-2x < 0$, the absolute value expression can be rewritten as:

$$-(5-2x) \leq 9$$

Simplifying the inequality gives us:

$$-5+2x \leq 9$$

Adding 5 to both sides of the inequality gives us:

$$2x \leq 14$$

Dividing both sides of the inequality by 2 gives us:

$$x \leq 7$$

Combining the Cases

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We have two cases: $x \geq -2$ and $x \leq 7$. However, we need to consider the original inequality $2|5-2x|-3 \leq 15$. When $x \geq -2$, the absolute value expression is non-negative, and when $x \leq 7$, the absolute value expression is negative.

Solution Set

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To find the solution set, we need to combine the two cases. However, we need to consider the intersection of the two cases. The intersection of $x \geq -2$ and $x \leq 7$ is $[-2, 7]$.

Conclusion

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In conclusion, the solution to the inequality $2|5-2x|-3 \leq 15$ is $[-2, 7]$. This solution set represents the values of $x$ that satisfy the original inequality.

Comparison with Answer Choices

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Let's compare the solution set $[-2, 7]$ with the answer choices:

• A. $(-\infty, -2) \cup (7, \infty)$: This solution set does not include the interval $[-2, 7]$.
• B. $(-\infty, 1.5) \cup (7.5, \infty)$: This solution set does not include the interval $[-2, 7]$.
• C. $[-2, 7]$: This solution set matches the solution set we obtained.
• D. $[1.5, 7.5]$: This solution set does not include the interval $[-2, 7]$.

Therefore, the correct answer is C. $[-2, 7]$.

Final Answer

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The final answer is $\boxed{C. [-2, 7]}$.

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Introduction

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In our previous article, we explored the concept of solving inequalities with absolute values. We walked through a step-by-step guide on how to solve the inequality $2|5-2x|-3 \leq 15$ and obtained the solution set $[-2, 7]$. In this article, we will provide a Q&A guide to help you better understand the concept and apply it to different types of inequalities.

Q&A

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Q: What is the first step in solving an inequality with absolute values?

A: The first step is to isolate the absolute value expression. This can be done by adding, subtracting, multiplying, or dividing both sides of the inequality by a constant.

Q: How do I determine the sign of the absolute value expression?

A: To determine the sign of the absolute value expression, you need to consider the two cases: when the expression inside the absolute value is non-negative and when it is negative. When the expression inside the absolute value is non-negative, the absolute value expression is equal to the expression itself. When the expression inside the absolute value is negative, the absolute value expression is equal to the negative of the expression.

Q: How do I rewrite the absolute value expression in the two cases?

A: In the case when the expression inside the absolute value is non-negative, you can rewrite the absolute value expression as the expression itself. In the case when the expression inside the absolute value is negative, you can rewrite the absolute value expression as the negative of the expression.

Q: How do I solve the inequality in the two cases?

A: In the case when the expression inside the absolute value is non-negative, you can solve the inequality by adding, subtracting, multiplying, or dividing both sides of the inequality by a constant. In the case when the expression inside the absolute value is negative, you can solve the inequality by adding, subtracting, multiplying, or dividing both sides of the inequality by a constant and then reversing the inequality sign.

Q: How do I combine the two cases to find the solution set?

A: To combine the two cases, you need to find the intersection of the two solution sets. This can be done by finding the values of the variable that satisfy both inequalities.

Q: What are some common mistakes to avoid when solving inequalities with absolute values?

A: Some common mistakes to avoid when solving inequalities with absolute values include:

• Not isolating the absolute value expression
• Not considering the two cases
• Not rewriting the absolute value expression correctly
• Not solving the inequality correctly in each case
• Not combining the two cases correctly to find the solution set

Example Problems

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Problem 1

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Solve the inequality $|x-3| \leq 2$.

Solution

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To solve the inequality, we need to isolate the absolute value expression. We can do this by adding 3 to both sides of the inequality:

$$|x-3+3| \leq 2+3$$

Simplifying the inequality gives us:

$$|x| \leq 5$$

We can rewrite the absolute value expression in the two cases:

• When $x \geq 0$, the absolute value expression is equal to the expression itself: $x \leq 5$
• When $x < 0$, the absolute value expression is equal to the negative of the expression: $-x \leq 5$

Solving the inequality in the two cases gives us:

• When $x \geq 0$, the solution set is $[0, 5]$
• When $x < 0$, the solution set is $[-5, 0)$

Combining the two cases, we find that the solution set is $[-5, 5]$.

Problem 2

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Solve the inequality $|2x-1| \geq 3$.

Solution

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To solve the inequality, we need to isolate the absolute value expression. We can do this by adding 1 to both sides of the inequality:

$$|2x-1+1| \geq 3+1$$

Simplifying the inequality gives us:

$$|2x| \geq 4$$

We can rewrite the absolute value expression in the two cases:

• When $x \geq 0$, the absolute value expression is equal to the expression itself: $2x \geq 4$
• When $x < 0$, the absolute value expression is equal to the negative of the expression: $-2x \geq 4$

Solving the inequality in the two cases gives us:

• When $x \geq 0$, the solution set is $[2, \infty)$
• When $x < 0$, the solution set is $(-\infty, -2)$

Combining the two cases, we find that the solution set is $(-\infty, -2) \cup [2, \infty)$.

Conclusion

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In conclusion, solving inequalities with absolute values requires a clear understanding of the concept and a step-by-step approach. By following the Q&A guide and example problems, you can develop the skills and confidence to tackle more complex inequalities. Remember to isolate the absolute value expression, consider the two cases, rewrite the absolute value expression correctly, solve the inequality correctly in each case, and combine the two cases correctly to find the solution set.