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Chapter 6: Problem 55

Solve each inequality and graph the solution set on a number line. \(\frac{x}{4}-\frac{3}{2} \leq \frac{x}{2}+1\)

Short Answer

Expert verified
The solution of inequality is \(x \geq -10\). We graph this by drawing a number line, marking -10 on it and shading the part of the number line that is to the right of -10 inclusive of -10.

Step by step solution

01

Multiply both sides of inequality by 4 to eliminate fractions

To eliminate the fractions, the entire equation has to be multiplied by 4, which gives: \(x-6 \leq 2x+4\).
02

Combine similar terms

To solve for \(x\), similar terms have to be put together. Therefore, subtract \(x\) and 4 from both sides: \(x-2x \leq 4+6\) which simplifies to \(-x \leq 10\).
03

Solve for \(x\)

To clear the negative sign before \(x\), one could multiply both sides of the inequality by -1. However, when multiplying by a negative, the inequality sign direction changes. So, multiplying the inequality \(-x \leq 10\) by -1 will give \(x \geq -10\).
04

Graph the solution on a number line

To graph the solution, draw a number line, and mark -10 on it. Since \(x\) is greater than or equal to -10, shade the part of the number line that is to the RIGHT of -10 inclusive of -10.

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