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Chapter 1: Problem 33

Use interval notation to express solution sets and graph each solution set on a number line. Solve each linear inequality. $$8 x-11 \leq 3 x-13$$

Short Answer

Expert verified
The solution to the inequality is \(x \leq -0.4\). In interval notation, this is expressed as \[(-\infty, -0.4]\]. The graph should be shaded to the left of -0.4, inclusively.

Step by step solution

01

Combine like terms

Subtract \(3x\) from both sides to collect all the terms containing x on one side: \[8x - 3x \leq 3x - 3x -13\] which simplifies to, \[5x - 11 \leq -13\]
02

Isolate the x term

Add 11 to both sides to isolate the x term: \(5x - 11 + 11 \leq -13 + 11\) which simplifies to, \[5x \leq -2\]
03

Solve for x

Divide both sides by 5 to solve for x: \(5x/5 \leq -2/5\), thus, \[x \leq -0.4\]
04

Write the solution in interval notation

The solution in interval notation is \[(-\infty, -0.4]\] inclusive of -0.4.
05

Graph the solution set

On the number line, shade all the numbers less than or equal to -0.4, and indicate -0.4 with a closed circle to show that -0.4 is included in the solution.

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