Q. 2 TB - Calculus

Question

Find the solution sets of each of the following inequalities.

$0 < |x + 5| < 0.1$

Step-by-Step Solution

Answer

The solution set of inequality is $\left(-5.1,-5\right)\cup \left(-5,-4.9\right).$

Step 1. Given information

Given inequality is $0 < |x + 5| < 0.1.$

Step 2. simplify the first inequality.

Take inequality $0 < |x + 5|.$

$$\begin{gathered} 0 < \left(x + 5\right) \\ -5<x \end{gathered}$$

or $$\begin{gathered} 0 <- \left(x + 5\right) \\ 0<-x+5 \\ -5>x \end{gathered}$$

So  $x\in \left(-\infty ,-5\right)\cup \left(-5,\infty \right)$

Step 3. simplify the second inequality.

Take inequality $|x + 5| < 0.1.$

$$\begin{gathered} |x + 5| < 0.1 \\ x + 5 < 0.1 \\ x<-4.9 \end{gathered}$$

or

$$\begin{gathered} |x + 5| < 0.1 \\ -\left(x + 5\right)< 0.1 \\ -x-5<0.1 \\ x>-5.1 \end{gathered}$$

so  $$\begin{gathered} -5.1<x<-4.9 \\ x\in \left(-5.1,-4.9\right) \end{gathered}$$

Step 4. The solution set of the inequality.

Combine both inequalities.

$$\begin{gathered} x\in \left(\left(-\infty ,-5\right)\cup \left(-5,\infty \right)\right)\cap \left(-5.1,-4.9\right) \\ x\in \left[\left(-\infty ,-5\right)\cap \left(-5.1,-4.9\right)\right]\cup \left[\left(-5,\infty \right)\cap \left(-5.1,-4.9\right)\right] \\ x\in \left(-5.1,-5\right)\cup \left(-5,-4.9\right) \end{gathered}$$