Which Of The Following Is The Solution To The Compound Inequality Below? ${ 7x + \frac{3}{4} \geq 13 }$ Or ${ \frac{5}{2}x - \frac{1}{3} \geq -\frac{11}{2} }$A. { X \geq \frac{343}{4} $}$ Or [$ X \leq

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Introduction

Compound inequalities are a combination of two or more inequalities joined by the words "and" or "or." They can be solved using various methods, including graphing, substitution, and elimination. In this article, we will focus on solving compound inequalities using the substitution method. We will also explore the concept of compound inequalities and provide examples to illustrate the solution process.

What are Compound Inequalities?

Compound inequalities are a combination of two or more inequalities joined by the words "and" or "or." They can be written in the following forms:

  • a≥ba \geq b or c≥dc \geq d
  • a≤ba \leq b and c≤dc \leq d
  • a≥ba \geq b and c≥dc \geq d

Compound inequalities can be solved using various methods, including graphing, substitution, and elimination.

Solving Compound Inequalities Using Substitution

To solve a compound inequality using substitution, we need to follow these steps:

  1. Simplify the inequality: Simplify the inequality by combining like terms and eliminating any fractions.
  2. Substitute the expression: Substitute the expression into the inequality.
  3. Solve the inequality: Solve the inequality using the substitution method.
  4. Graph the solution: Graph the solution on a number line.

Example 1: Solving a Compound Inequality Using Substitution

Let's consider the following compound inequality:

{ 7x + \frac{3}{4} \geq 13 \}$ or ${ \frac{5}{2}x - \frac{1}{3} \geq -\frac{11}{2} \}

To solve this compound inequality, we need to follow the steps outlined above.

Step 1: Simplify the Inequality

First, we need to simplify the inequality by combining like terms and eliminating any fractions.

{ 7x + \frac{3}{4} \geq 13 \}

Multiply both sides of the inequality by 4 to eliminate the fraction:

{ 28x + 3 \geq 52 \}

Subtract 3 from both sides of the inequality:

{ 28x \geq 49 \}

Divide both sides of the inequality by 28:

{ x \geq \frac{49}{28} \}

{ x \geq \frac{7}{4} \}

Step 2: Substitute the Expression

Next, we need to substitute the expression into the inequality.

{ \frac{5}{2}x - \frac{1}{3} \geq -\frac{11}{2} \}

Multiply both sides of the inequality by 6 to eliminate the fractions:

{ 15x - 2 \geq -33 \}

Add 2 to both sides of the inequality:

{ 15x \geq -31 \}

Divide both sides of the inequality by 15:

{ x \geq -\frac{31}{15} \}

Step 3: Solve the Inequality

Now, we need to solve the inequality using the substitution method.

We have two inequalities:

{ x \geq \frac{7}{4} \}

{ x \geq -\frac{31}{15} \}

Since the compound inequality is "or," we need to find the union of the two inequalities.

The solution to the compound inequality is:

{ x \geq \frac{7}{4} }$ or ${ x \geq -\frac{31}{15} \}

Step 4: Graph the Solution

Finally, we need to graph the solution on a number line.

The solution to the compound inequality is:

{ x \geq \frac{7}{4} }$ or ${ x \geq -\frac{31}{15} \}

The graph of the solution is a number line with two intervals: ${ \frac{7}{4}, \infty }$ and ${ -\frac{31}{15}, \infty }$.

Conclusion

In this article, we have discussed the concept of compound inequalities and provided a step-by-step guide on how to solve them using the substitution method. We have also explored the concept of compound inequalities and provided examples to illustrate the solution process. By following the steps outlined in this article, you can solve compound inequalities with ease.

Frequently Asked Questions

Q: What is a compound inequality?

A: A compound inequality is a combination of two or more inequalities joined by the words "and" or "or."

Q: How do I solve a compound inequality?

A: To solve a compound inequality, you need to follow the steps outlined in this article: simplify the inequality, substitute the expression, solve the inequality, and graph the solution.

Q: What is the solution to the compound inequality ${ 7x + \frac{3}{4} \geq 13 }$ or ${ \frac{5}{2}x - \frac{1}{3} \geq -\frac{11}{2} }$?

A: The solution to the compound inequality is ${ x \geq \frac{343}{4} }$ or ${ x \leq -\frac{31}{15} }$.

Q: How do I graph the solution to a compound inequality?

A: To graph the solution to a compound inequality, you need to graph the solution on a number line. The solution is a number line with two intervals: ${ \frac{7}{4}, \infty }$ and ${ -\frac{31}{15}, \infty }$.

References

  • "Algebra and Trigonometry" by Michael Sullivan
  • "College Algebra" by James Stewart
  • "Mathematics for the Nonmathematician" by Morris Kline

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Introduction

Compound inequalities are a combination of two or more inequalities joined by the words "and" or "or." They can be solved using various methods, including graphing, substitution, and elimination. In this article, we will provide a comprehensive Q&A guide on compound inequalities, covering topics such as solving compound inequalities, graphing solutions, and more.

Q&A: Solving Compound Inequalities

Q: What is the first step in solving a compound inequality?

A: The first step in solving a compound inequality is to simplify the inequality by combining like terms and eliminating any fractions.

Q: How do I simplify a compound inequality?

A: To simplify a compound inequality, you need to follow these steps:

  1. Combine like terms.
  2. Eliminate any fractions by multiplying both sides of the inequality by the least common multiple (LCM) of the denominators.
  3. Simplify the inequality.

Q: What is the next step in solving a compound inequality?

A: The next step in solving a compound inequality is to substitute the expression into the inequality.

Q: How do I substitute the expression into the inequality?

A: To substitute the expression into the inequality, you need to follow these steps:

  1. Identify the expression to be substituted.
  2. Substitute the expression into the inequality.
  3. Simplify the inequality.

Q: What is the final step in solving a compound inequality?

A: The final step in solving a compound inequality is to graph the solution on a number line.

Q: How do I graph the solution to a compound inequality?

A: To graph the solution to a compound inequality, you need to follow these steps:

  1. Identify the solution to the compound inequality.
  2. Graph the solution on a number line.
  3. Label the intervals on the number line.

Q&A: Graphing Solutions

Q: What is the purpose of graphing the solution to a compound inequality?

A: The purpose of graphing the solution to a compound inequality is to visualize the solution and identify the intervals that satisfy the inequality.

Q: How do I graph the solution to a compound inequality?

A: To graph the solution to a compound inequality, you need to follow these steps:

  1. Identify the solution to the compound inequality.
  2. Graph the solution on a number line.
  3. Label the intervals on the number line.

Q: What are the different types of intervals that can be graphed on a number line?

A: The different types of intervals that can be graphed on a number line are:

  • Open intervals: ${ a, b )$
  • Closed intervals: \[ a, b }$
  • Half-open intervals: ${ a, b }$ and ${ a, b }$
  • Half-closed intervals: ${ a, b }$ and ${ a, b }$

Q&A: Common Mistakes

Q: What are some common mistakes to avoid when solving compound inequalities?

A: Some common mistakes to avoid when solving compound inequalities are:

  • Not simplifying the inequality before substituting the expression.
  • Not identifying the correct intervals to graph on the number line.
  • Not labeling the intervals on the number line.

Q: How can I avoid making these mistakes?

A: To avoid making these mistakes, you need to:

  • Carefully simplify the inequality before substituting the expression.
  • Identify the correct intervals to graph on the number line.
  • Label the intervals on the number line.

Conclusion

In this article, we have provided a comprehensive Q&A guide on compound inequalities, covering topics such as solving compound inequalities, graphing solutions, and common mistakes to avoid. By following the steps outlined in this article, you can solve compound inequalities with ease and avoid common mistakes.

Frequently Asked Questions

Q: What is a compound inequality?

A: A compound inequality is a combination of two or more inequalities joined by the words "and" or "or."

Q: How do I solve a compound inequality?

A: To solve a compound inequality, you need to follow the steps outlined in this article: simplify the inequality, substitute the expression, solve the inequality, and graph the solution.

Q: What is the solution to the compound inequality ${ 7x + \frac{3}{4} \geq 13 }$ or ${ \frac{5}{2}x - \frac{1}{3} \geq -\frac{11}{2} }$?

A: The solution to the compound inequality is ${ x \geq \frac{343}{4} }$ or ${ x \leq -\frac{31}{15} }$.

Q: How do I graph the solution to a compound inequality?

A: To graph the solution to a compound inequality, you need to graph the solution on a number line. The solution is a number line with two intervals: ${ \frac{7}{4}, \infty }$ and ${ -\frac{31}{15}, \infty }$.

References

  • "Algebra and Trigonometry" by Michael Sullivan
  • "College Algebra" by James Stewart
  • "Mathematics for the Nonmathematician" by Morris Kline