Rewriting the Compound Inequality

Oct 13, 2025 by wbrum.org 136 views

Understanding Absolute Value Inequalities

Absolute value inequalities are a type of mathematical expression that involves the absolute value of a variable or expression. They are used to represent a range of values that a variable or expression can take. In this article, we will focus on solving absolute value inequalities, specifically the compound inequality ${-7 \leq 8-3q \leq 7}$ .

The Compound Inequality

The compound inequality ${-7 \leq 8-3q \leq 7}$ is a statement that says the value of ${8-3q}$ is between ${-7}$ and ${7}$ , inclusive. This means that the value of ${8-3q}$ can be any real number between ${-7}$ and ${7}$ .

Rewriting the Compound Inequality

To solve the compound inequality, we need to rewrite it as a single absolute value inequality. We can do this by combining the two parts of the compound inequality into a single absolute value expression.

Step 1: Isolate the Absolute Value Expression

The first step in rewriting the compound inequality is to isolate the absolute value expression. We can do this by subtracting ${8}$ from all three parts of the inequality.

${-7 - 8 \leq 8 - 3q - 8 \leq 7 - 8}$

This simplifies to:

${-15 \leq -3q \leq -1}$

Step 2: Divide by -3

The next step is to divide all three parts of the inequality by ${-3}$ . When we divide by a negative number, we need to reverse the direction of the inequality signs.

${\frac{-15}{-3} \geq \frac{-3q}{-3} \geq \frac{-1}{-3}}$

This simplifies to:

${5 \geq q \geq \frac{1}{3}}$

Step 3: Write the Absolute Value Inequality

Now that we have isolated the absolute value expression and divided by ${-3}$ , we can write the absolute value inequality.

${|q - 5| \geq \frac{4}{3}}$

Step 4: Solve the Absolute Value Inequality

To solve the absolute value inequality, we need to consider two cases: when the expression inside the absolute value is positive, and when it is negative.

Case 1: When the expression inside the absolute value is positive

When the expression inside the absolute value is positive, we can write the inequality as:

${q - 5 \geq \frac{4}{3}}$

Solving for ${q}$ , we get:

${q \geq 5 + \frac{4}{3}}$

${q \geq \frac{19}{3}}$

Case 2: When the expression inside the absolute value is negative

When the expression inside the absolute value is negative, we can write the inequality as:

${q - 5 \leq -\frac{4}{3}}$

Solving for ${q}$ , we get:

${q \leq 5 - \frac{4}{3}}$

${q \leq \frac{11}{3}}$

Combining the Two Cases

To find the final solution, we need to combine the two cases. The solution is the union of the two cases.

${q \geq \frac{19}{3} \text{ or } q \leq \frac{11}{3}}$

Conclusion

In this article, we have solved the compound inequality ${-7 \leq 8-3q \leq 7}$ by rewriting it as a single absolute value inequality. We have isolated the absolute value expression, divided by ${-3}$ , and written the absolute value inequality. We have then solved the absolute value inequality by considering two cases: when the expression inside the absolute value is positive, and when it is negative. The final solution is the union of the two cases.

Answer

The correct answer is:

A. ${|8-3a| \geq 7}$

Q: What is an absolute value inequality?

A: An absolute value inequality is a type of mathematical expression that involves the absolute value of a variable or expression. It is used to represent a range of values that a variable or expression can take.

Q: How do I solve an absolute value inequality?

A: To solve an absolute value inequality, you need to follow these steps:

Isolate the absolute value expression.
Divide by the coefficient of the variable (if it is not 1).
Write the absolute value inequality.
Solve the absolute value inequality by considering two cases: when the expression inside the absolute value is positive, and when it is negative.

Q: What is the difference between an absolute value inequality and a compound inequality?

A: An absolute value inequality is a type of inequality that involves the absolute value of a variable or expression. A compound inequality is a type of inequality that involves two or more inequalities joined by "and" or "or". For example, ${-7 \leq 8-3q \leq 7}$ is a compound inequality, while ${|8-3q| \geq 7}$ is an absolute value inequality.

Q: How do I rewrite a compound inequality as an absolute value inequality?

A: To rewrite a compound inequality as an absolute value inequality, you need to follow these steps:

Isolate the absolute value expression.
Divide by the coefficient of the variable (if it is not 1).
Write the absolute value inequality.

Q: What is the significance of the absolute value in an absolute value inequality?

A: The absolute value in an absolute value inequality represents the distance of the variable or expression from zero. It is used to represent a range of values that a variable or expression can take.

Q: How do I solve an absolute value inequality with a negative coefficient?

A: To solve an absolute value inequality with a negative coefficient, you need to follow these steps:

Isolate the absolute value expression.
Divide by the coefficient of the variable (if it is not 1).
Write the absolute value inequality.
Solve the absolute value inequality by considering two cases: when the expression inside the absolute value is positive, and when it is negative.

Q: What is the difference between an absolute value inequality and a linear inequality?

A: An absolute value inequality is a type of inequality that involves the absolute value of a variable or expression. A linear inequality is a type of inequality that involves a linear expression. For example, ${|8-3q| \geq 7}$ is an absolute value inequality, while ${8-3q \geq 7}$ is a linear inequality.

Q: How do I graph an absolute value inequality on a number line?

A: To graph an absolute value inequality on a number line, you need to follow these steps:

Find the critical points of the inequality.
Plot the critical points on the number line.
Shade the region that satisfies the inequality.

Q: What is the significance of the critical points in an absolute value inequality?

A: The critical points in an absolute value inequality represent the values of the variable or expression that make the inequality true. They are used to graph the inequality on a number line.

Conclusion

In this article, we have answered some common questions about solving absolute value inequalities. We have covered topics such as the definition of an absolute value inequality, how to solve an absolute value inequality, and how to graph an absolute value inequality on a number line. We hope that this article has been helpful in understanding absolute value inequalities.