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Chapter 1: Problem 108

When 4 times a number is subtracted from \(5,\) the absolute value of the difference is at most \(13 .\) Use interval notation to express the set of all numbers that satisfy this condition.

Short Answer

Expert verified
The set of all numbers that satisfy the given condition is \(-2 \leq x \leq 4.5\), which in interval notation is expressed as \([-2, 4.5]\).

Step by step solution

01

Express the Problem as an Inequality

The first step is to translate the problem into a mathematical inequality. We know that absolute value of the difference between 5 and 4 times a number \(x\) is at most \(13\). This can be written as \( |5 - 4x| \leq 13 \).
02

Solve the Inequality

The next step is to solve the inequality. The absolute value inequality can be split into two separate inequalities: \(5 - 4x \leq 13\) and \(5 - 4x \geq -13\).
03

Solve the First Inequality

To solve the first inequality \(5 - 4x \leq 13\), we subtract 5 from both sides to isolate the term with \(x\): \(-4x \leq 8\). We then divide both sides by -4, which flips the direction of the inequality, to get \(x \geq -2\).
04

Solve the Second Inequality

We perform the same steps to solve the second inequality \(5 - 4x \geq -13\). Subtracting 5 from both sides gives \(-4x \geq -18\). After dividing by -4 and flipping the inequality, we get \(x \leq 4.5\).
05

Combine the Results

The final step is to take the intersection of the solutions for both inequalities. The number \(x\) has to satisfy both \(x \geq -2\) and \(x \leq 4.5\), which means the solution is \(-2 \leq x \leq 4.5\).

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Most popular questions from this chapter

Describe ways in which solving a linear inequality is similar to solving a linear equation.

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