Section A.6 – Linear Inequalities
- The properties of inequalities
- Solving linear inequalities
- Solving absolute value inequalities
- Writing the solution in interval notation
- Graphing the solution set
Exercises
Directions: You should try to solve each problem first, and then click "Reveal Answer" to check your answer. You can click "Watch Video" if you need help with a problem.
1. Solve the following inequality. Graph its solution set. \(\dfrac{x}{3}+\dfrac{1}{2}> \dfrac{4x-1}{6}\)
Answer: \(x<2\)
Solution:
\(
\begin{align}
6\cdot\left(\dfrac{x}{3}+\dfrac{1}{2}\right)&>6\cdot\left(\dfrac{4x-1}{6}\right)\\
2x + 3 &> 4x-1\\
4&>2x\\
2&>x
\end{align}\)
Solution:
\(
\begin{align}
6\cdot\left(\dfrac{x}{3}+\dfrac{1}{2}\right)&>6\cdot\left(\dfrac{4x-1}{6}\right)\\
2x + 3 &> 4x-1\\
4&>2x\\
2&>x
\end{align}\)
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: \[\begin{align}
2\cdot(-3)&<2\cdot\left(\dfrac{2x-1}{2}\right)\leq2\cdot4\\
-6&<2x-1\leq8\\
-5&<2x\leq9\\
-\dfrac{5}{2}&\leq x \leq \dfrac{9}{2}
\end{align}\]
Solution: \[\begin{align}
2\cdot(-3)&<2\cdot\left(\dfrac{2x-1}{2}\right)\leq2\cdot4\\
-6&<2x-1\leq8\\
-5&<2x\leq9\\
-\dfrac{5}{2}&\leq x \leq \dfrac{9}{2}
\end{align}\]
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:
\(\begin{alignat}
4x-5 &\geq 11 \qquad &&\textrm{OR} \qquad 4x-5 &&\leq -11\\
4x &\geq 16 \qquad &&\textrm{OR} \qquad \qquad 4x &&\leq -6\\
x&\geq 4 \qquad &&\textrm{OR} \qquad \qquad x &&\leq-\dfrac{3}{2}
\end{alignat}\)
Solution:
\(\begin{alignat}
4x-5 &\geq 11 \qquad &&\textrm{OR} \qquad 4x-5 &&\leq -11\\
4x &\geq 16 \qquad &&\textrm{OR} \qquad \qquad 4x &&\leq -6\\
x&\geq 4 \qquad &&\textrm{OR} \qquad \qquad x &&\leq-\dfrac{3}{2}
\end{alignat}\)