# Section A.6 – Linear Inequalities

Section Details.
• The properties of inequalities
• Solving linear inequalities
• Solving absolute value inequalities
• Writing the solution in interval notation
• Graphing the solution set

### Practice Problems

Directions. The following are review problems for the section. We recommend you work the problems yourself, and then click "Answer" to check your answer. If you do not understand a problem, you can click "Video" to learn how to solve it.
1. Solve the following inequality. Graph its solution set. $$\dfrac{x}{3}+\dfrac{1}{2}> \dfrac{4x-1}{6}$$

Answer: $$x<2$$

Solution:

\begin{align} 6\cdot\left(\dfrac{x}{3}+\dfrac{1}{2}\right)&>6\cdot\left(\dfrac{4x-1}{6}\right)\\ 2x + 3 &> 4x-1\\ 4&>2x\\ 2&>x \end{align}

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2. Solve the following inequality. Graph its solution set. $$-3<\dfrac{2x-1}{2}\leq 4$$

Answer: $$-\dfrac{5}{2}<x\leq\dfrac{9}{2}$$

Solution:
\begin{align} 2\cdot(-3)&<2\cdot\left(\dfrac{2x-1}{2}\right)\leq2\cdot4\\ -6&<2x-1\leq8\\ -5&<2x\leq9\\ -\dfrac{5}{2}&\leq x \leq \dfrac{9}{2} \end{align}

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3. Solve the following inequality. Graph its solution set. $$|4x-5|\geq11$$

Answer: $$x\leq-\dfrac{3}{2}\quad \text{ OR } \quad x\geq4$$

Solution:
\begin{alignat} 4x-5 &\geq 11 \qquad &&\textrm{OR} \qquad 4x-5 &&\leq -11\\ 4x &\geq 16 \qquad &&\textrm{OR} \qquad \qquad 4x &&\leq -6\\ x&\geq 4 \qquad &&\textrm{OR} \qquad \qquad x &&\leq-\dfrac{3}{2} \end{alignat}

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4. Solve the following inequality. $$|9-2x|-2\leq -1$$

$$[4,5]$$

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