Q. 15 - 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 $0<|x-0|<0.75,$then $\left|x^2-0\right|<0.5.$

Step-by-Step Solution

Answer

 The implication is false. 

Step 1. Given information.

The given statement is the following. 

If $0<|x-0|<0.75$ then $\left|x^2-0\right|<0.5.$

Step 2. Justification.

Substitute $x=-0.72$in first inequality.

$$\begin{gathered} 0<|x-0|<0.75 \\ 0\stackrel{?}{<}|-0.72-0|\stackrel{?}{<}0.75 \\ 0<0.72<0.75 \end{gathered}$$

Substitute $x=-0.72$in second inequality.

$$\begin{gathered} \left|x^2-0\right|<0.5 \\ \left|{(-0.72)}^2-0\right|\stackrel{?}{<}0.5 \\ 0.5184\nless 0.5 \end{gathered}$$

The first inequality is satisfied at $x=-0.72$but the second inequality is not satisfied.

So the implication is false.