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Chapter 1: Problem 37

Solve each inequality. Give the solution set using interval notation. $$|5-3 x| \leq 7$$

Short Answer

Expert verified
The solution set is \([-\frac{2}{3}, 4]\).

Step by step solution

01

- Understand the absolute value inequality

The inequality \( |5 - 3x| \leq 7 \) means that the expression inside the absolute value can satisfy two conditions: \( 5 - 3x \leq 7 \) and \( 5 - 3x \geq -7 \).
02

- Solve the first condition

First solve the inequality \( 5 - 3x \leq 7 \). Subtract 5 from both sides to get \( -3x \leq 2 \). Then, divide by -3 and reverse the inequality sign to find \( x \geq -\frac{2}{3} \).
03

- Solve the second condition

Next, solve the inequality \( 5 - 3x \geq -7 \). Subtract 5 from both sides to get \( -3x \geq -12 \). Then, divide by -3 and reverse the inequality sign to find \( x \leq 4 \).
04

- Combine the two inequalities

Combine the results from Step 2 (\( x \geq -\frac{2}{3} \)) and Step 3 (\( x \leq 4 \)) to get the solution set \( -\frac{2}{3} \leq x \leq 4 \).
05

- Write the solution in interval notation

The solution set in interval notation is \([-\frac{2}{3}, 4]\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Solving Inequalities
Solving inequalities involves finding the values of a variable that make the inequality true. Unlike equations, inequalities show a range of possible solutions, not just one specific value. To solve inequalities, follow these key steps:
  • Isolate the variable on one side of the inequality.
  • Perform operations like addition, subtraction, multiplication, or division on both sides.
  • If you multiply or divide by a negative number, remember to reverse the inequality sign.
Let's look at the solution for the inequality from the exercise:
The original inequality is: \( |5 - 3x| \leq 7 \)
We're dealing with an absolute value, so we will have two scenarios to consider.
Interval Notation
Once we solve an inequality, we often express the solution set using interval notation. Interval notation is a concise way to write the range of values for the variable that satisfy the inequality. Here's how it works:
  • Use square brackets \[ ... \] to include endpoints.
  • Use parentheses \( ... \) to exclude endpoints.
In the solved example, we find that \(-\frac{2}{3} \leq x \leq 4\). In interval notation, this is written as \([\frac{-2}{3}, 4]\). This means that the solutions include all numbers between -2/3 and 4, including both -2/3 and 4 themselves.
Absolute Value
Absolute value represents the distance of a number from zero on the number line, regardless of direction. It is always non-negative. The absolute value of a number \x\ is written as \|x|\ and defined as follows:
\[ |x| = \begin{cases} x & \text{if } x \geq 0 \ -x & \text{if } x < 0 \ \end{cases} \] Absolute value inequalities represent ranges and can typically be broken into two separate inequalities to solve. For example, solving \|5 - 3x|\results in solving both \( 5 - 3x \leq 7 \) and \( 5 - 3x \geq -7 \). This allows you to find all possible values of x that make the inequality true, combining these to get a comprehensive solution set in interval notation.

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Most popular questions from this chapter

Solve each rational inequality. Write each solution set in interval notation. $$\frac{x+3}{x-5} \leq 1$$$

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