Chapter 12.3, Problem 3MRE

To determine

To calculate: The solution set for open sentence $\left|3p+5\right|<7$ and graph it.

Answer to Problem 3MRE

The solution set is $\left(-4,\frac{2}{3}\right)$ . The graph is provided below,

Explanation of Solution

Given information:

The open sentence $\left|3p+5\right|<7$ .

Formula used:

For an algebraic expression w and a real number n such that $n>0$ , absolute value inequality is $\left|w\right|<n$ if and only if $-n<w<n$ .

Calculation:

Consider the open sentence $\left|3p+5\right|<7$ .

Recall that for an algebraic expression w and a real number n such that $n>0$ , absolute value inequality is $\left|w\right|<n$ if and only if $-n<w<n$ .

Apply it,

  $-7<3p+5<7$

Subtract 5 in each part of the above inequality,

  $\begin{array}{c}-7-5<3p+5-5<7-5\\ -12<3p<2\end{array}$

Divide each part of above inequality by 3,

  $\begin{array}{c}-7-5<3p+5-5<7-5\\ -12<3p<2\\ -\frac{12}{3}<\frac{3p}{3}<\frac{2}{3}\\ -4<p<\frac{2}{3}\end{array}$

Write it in the interval form $\left(-4,\frac{2}{3}\right)$ .

The solution set is the open interval that is all values of p that lie between $-4\text{ and }\frac{2}{3}$ .

On the real number mark the points $p=-4\text{ and }p=\frac{2}{3}$ and darken the line between the points and the end points are not included.

Thus, the solution set is $\left(-4,\frac{2}{3}\right)$ .