Q. 16 - 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$0<|x+2|<0.1,$then $\left|x^2-4\right|<0.4.$

Step-by-Step Solution

Answer

The implication is false.  

Step 1. Given information.

The given statement is the following. 

If $0<|x+2|<0.1,$then $\left|x^2-4\right|<0.4.$

Step 2. Justification.

Substitute $x=-1.9$in first inequality.

$$\begin{gathered} 0<|x+2|<0.1 \\ 0<|-1.9+2|<0.1 \\ 0<0.1\nless 0.1 \end{gathered}$$

substitute $x=-1.9$in second inequality.

$$\begin{gathered} \left|x^2-4\right|<0.4 \\ \left|1.9^2-4\right|\stackrel{?}{<}0.4 \\ 0.39<0.4 \end{gathered}$$

The second inequality was satisfied but the first inequality did not satisfied. 

So the implication is false.