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


    To see the full video page and find related videos, click the following link.
    WIR1 20B M150 V4

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


    To see the full video page and find related videos, click the following link.
    WIR1 20B M150 V5

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


    To see the full video page and find related videos, click the following link.
    WIR1 20B M150 V6

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

    \([4,5]\)


    To see the full video page and find related videos, click the following link.
    WIR8 20B M150 V06