Q. 13 - 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 $0<|x-2|<1,$then $\left|x^2-4\right|<0.5.$

Step-by-Step Solution

Answer

The implication is false.

Step 1. Given information.

The given statement is the following.

If $0<|x-2|<1,$then $\left|x^2-4\right|<0.5.$

Step 2. Explanation.

Substitute $x=2.5$in first inequality.

$$\begin{gathered} 0<|x-2|<1 \\ 0\stackrel{?}{<}|2.5-2|\stackrel{?}{<}1 \\ 0\stackrel{?}{<}|0.5|\stackrel{?}{<}1 \\ 0<0.5<1 \end{gathered}$$

Substitute $x=2.5$in second inequality.

$$\begin{gathered} \left|x^2-4\right|<0.5 \\ \left|2.5^2-4\right|\stackrel{?}{<}0.5 \\ |2.25|\stackrel{?}{<}0.5 \\ 2.25\nless 0.5 \end{gathered}$$

The first inequality is satisfied with $x=2.5$but the second inequality did not satisfied.

So the implication is false