Q. 1 TB - Calculus

Question

Find the solution sets of each of the following inequalities.

$0 < |x - 2| < 0.5$

Step-by-Step Solution

Answer

The solution set of inequality is $\left(1.5,2\right)\cup \left(2,2.5\right).$

Step 1. Given information

Given inequality is $0 < |x - 2| < 0.5.$

Step 2. simplify the first inequality.

Take inequality $0 < |x - 2| .$

$$\begin{gathered} 0 < |x - 2| \\ 0 < x - 2 \\ 2<x \end{gathered}$$

or

$$\begin{gathered} 0 <- \left(x - 2\right) \\ 2>x \end{gathered}$$

so $x\in \left(-\infty ,2\right)\cup \left(2,\infty \right)$

Step 3. simplify the second inequality.

Take inequality $|x - 2| < 0.5$

$$\begin{gathered} |x - 2| < 0.5 \\ x - 2< 0.5 \\ x<2.5 \end{gathered}$$

or

$$\begin{gathered} |x - 2| < 0.5 \\ -\left(x - 2\right)< 0.5 \\ -x + 2< 0.5 \\ x>1.5 \end{gathered}$$

so $x\in \left(1.5,2.5\right)$

Step 4. The solution set of the inequality.

Combine solutions of both inequality

$$\begin{gathered} x\in \left[\left(-\infty ,2\right)\cup \left(2,\infty \right)\right]\cap \left(1.5,2.5\right) \\ x\in \left[\left(-\infty ,2\right)\cap \left(1.5,2.5\right)\right]\cup \left[\left(2,\infty \right)\cap \left(1.5,2.5\right)\right] \\ x\in \left(1.5,2\right)\cup \left(2,2.5\right) \end{gathered}$$