Q. 14 - Calculus

Question

Determine whether each implication that follows is true or

false. Use graphs to justify any implications that are true and

counterexamples for any implications that are false.

If $0<|x-2|<0.2,$then $\left|x^2-4\right|<1.$

Step-by-Step Solution

Answer

 The implication is true. 

Step 1. Given information.

The given statement is the following. 

If $0<|x-2|<0.2,$then $\left|x^2-4\right|<1.$

Step 2. Justification.

Substitute $x=1.9$in first inequality.

$$\begin{gathered} 0<|x-2|<0.2 \\ 0\stackrel{?}{<}|1.9-2|\stackrel{?}{<}0.2 \\ 0\stackrel{?}{<}|0.1|\stackrel{?}{<}0.2 \\ 0<0.1<0.2 \end{gathered}$$

Substitute $x=1.9$in second inequality.

$$\begin{gathered} \left|x^2-4\right|<1 \\ \left|1.9^2-4\right|\stackrel{?}{<}1 \\ \left|-0.39\right|\stackrel{?}{<}1 \\ 0.39<1 \end{gathered}$$

both inequalities are satisfied at $x=1.9$.

so the implication is true.

Step 3. Graph of inequalities.

Plot the graph of both inequalities.The graph 

Inequality$0<|x-2|<0.2$ is within inequality $\left|x^2-4\right|<1.$

so the implication is true.