Solve $|x-5| \leq 7$.1. State Your Answer As A Compound Inequality $A \leq X \leq B$: $\boxed{}$2. State Your Answer In Interval Notation \[A, B\]: $\boxed{}$

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Introduction

Absolute value inequalities are a type of mathematical problem that involves finding the solution set for an absolute value expression. In this article, we will focus on solving the absolute value inequality $|x-5| \leq 7$. We will first state the solution as a compound inequality and then express it in interval notation.

Step 1: Understand the Absolute Value Inequality

The given absolute value inequality is $|x-5| \leq 7$. This means that the distance between $x$ and $5$ is less than or equal to $7$. To solve this inequality, we need to find all the values of $x$ that satisfy this condition.

Step 2: Solve the Absolute Value Inequality

To solve the absolute value inequality, we can use the following steps:

• If $x-5 \geq 0$, then $|x-5| = x-5$.
• If $x-5 < 0$, then $|x-5| = -(x-5)$.

We can rewrite the inequality as:

• $x-5 \leq 7$ if $x-5 \geq 0$
• $-(x-5) \leq 7$ if $x-5 < 0$

Step 3: Solve the Inequality for $x-5 \geq 0$

If $x-5 \geq 0$, then we can add $5$ to both sides of the inequality to get:

$x \leq 12$

Step 4: Solve the Inequality for $x-5 < 0$

If $x-5 < 0$, then we can add $5$ to both sides of the inequality to get:

$-x+5 \leq 7$

Subtracting $5$ from both sides gives:

$-x \leq 2$

Multiplying both sides by $-1$ gives:

$x \geq -2$

Step 5: Combine the Solutions

Since the inequality $x-5 \geq 0$ is true when $x \geq 5$, we can combine the solutions as follows:

• If $x \geq 5$, then $x \leq 12$
• If $x < 5$, then $x \geq -2$

We can express this as a compound inequality:

$-2 \leq x \leq 12$

Step 6: Express the Solution in Interval Notation

The solution can be expressed in interval notation as:

$[-2, 12]$

Conclusion

In this article, we solved the absolute value inequality $|x-5| \leq 7$ and expressed the solution as a compound inequality and in interval notation. The solution is $-2 \leq x \leq 12$ and can be expressed in interval notation as $[-2, 12]$.

Final Answer

1. State your answer as a compound inequality $A \leq x \leq B$: $\boxed{-2 \leq x \leq 12}$
2. State your answer in interval notation ${A, B}$: $\boxed{[-2, 12]}$
   

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Introduction

In our previous article, we solved the absolute value inequality $|x-5| \leq 7$ and expressed the solution as a compound inequality and in interval notation. In this article, we will answer some frequently asked questions related to absolute value inequalities.

Q&A

Q1: What is an absolute value inequality?

A1: An absolute value inequality is a mathematical problem that involves finding the solution set for an absolute value expression. It is a type of inequality that involves the absolute value of a variable or expression.

Q2: How do I solve an absolute value inequality?

A2: To solve an absolute value inequality, you need to follow these steps:

• If $x-5 \geq 0$, then $|x-5| = x-5$.
• If $x-5 < 0$, then $|x-5| = -(x-5)$.
• Rewrite the inequality as two separate inequalities.
• Solve each inequality separately.
• Combine the solutions to get the final answer.

Q3: What is the difference between an absolute value inequality and a linear inequality?

A3: 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, $|x-5| \leq 7$ is an absolute value inequality, while $x \leq 12$ is a linear inequality.

Q4: How do I express the solution to an absolute value inequality in interval notation?

A4: To express the solution to an absolute value inequality in interval notation, you need to follow these steps:

• Identify the values of $x$ that satisfy the inequality.
• Write the values in order from smallest to largest.
• Use square brackets to indicate that the values are included in the solution set.
• Use parentheses to indicate that the values are not included in the solution set.

Q5: Can I use a calculator to solve an absolute value inequality?

A5: Yes, you can use a calculator to solve an absolute value inequality. However, you need to make sure that the calculator is set to the correct mode and that you are using the correct function.

Q6: How do I check my answer to an absolute value inequality?

A6: To check your answer to an absolute value inequality, you need to follow these steps:

• Plug in a value of $x$ that is in the solution set.
• Check that the value satisfies the original inequality.
• Plug in a value of $x$ that is not in the solution set.
• Check that the value does not satisfy the original inequality.

Q7: Can I use absolute value inequalities to solve real-world problems?

A7: Yes, you can use absolute value inequalities to solve real-world problems. For example, you can use absolute value inequalities to model the distance between two points, the cost of a product, or the time it takes to complete a task.

Q8: How do I graph an absolute value inequality on a number line?

A8: To graph an absolute value inequality on a number line, you need to follow these steps:

• Identify the values of $x$ that satisfy the inequality.
• Plot the values on a number line.
• Use a closed circle to indicate that the values are included in the solution set.
• Use an open circle to indicate that the values are not included in the solution set.

Q9: Can I use absolute value inequalities to solve systems of equations?

A9: Yes, you can use absolute value inequalities to solve systems of equations. For example, you can use absolute value inequalities to solve a system of linear equations or a system of nonlinear equations.

Q10: How do I use absolute value inequalities to model real-world problems?

A10: To use absolute value inequalities to model real-world problems, you need to follow these steps:

• Identify the variables and constants in the problem.
• Write an equation or inequality that models the problem.
• Solve the equation or inequality to get the solution.
• Interpret the solution in the context of the problem.

Final Answer

We hope that this Q&A article has helped you to understand absolute value inequalities and how to solve them. If you have any further questions, please don't hesitate to ask.