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Chapter 1: Problem 34

Use interval notation to express solution sets and graph each solution set on a number line. Solve each linear inequality. $$18 x+45 \leq 12 x-8$$

Short Answer

Expert verified
The solution to the inequality \(18x + 45 \leq 12x -8\) is \( x \leq -53/6\), expressed in interval notation as \([-鈭�, -53/6]\). The graph would start at -53/6 with a closed circle and shade to the left.

Step by step solution

01

Simplify the Inequality

Subtract 12x from both sides to isolate the x variable. This will result in: \(6x + 45 \leq -8\)
02

Further Isolate x

In order to completely isolate x, it's important to get rid of '+45' on its side. Therefore, subtract 45 from both sides. This results in: \(6x \leq -53\)
03

Solving for x

Now that x is isolated on one side, divide both sides by 6 to find the final solution: \(x \leq -53/6\) or in decimal form: \(x \leq -8.83\)
04

Express in Interval Notation

The values x can take on are less than or equal to -53/6. This is expressed in interval notation as: \([-鈭�, -53/6]\)
05

Graph on a Number Line

Draw a number line and mark -53/6. Draw a closed circle (indicating 'less than or equal to') at this point and shade to the left (indicating all numbers less than -53/6).

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