Question

Solve each inequality. Graph the solution set on number line.

$3p+1\le 7\text{or}2p-9\ge 7$

Short answer

Solution of inequality is $p\le 2\text{or}p\ge 8$

Solution set on number line is

Step-by-step solution

- Write properties of inequality

Properties of inequalities are written in following table:

- Simplify 3p+1≤7

To simplify given inequality, use subtraction and division property of inequalities. Subtract by same number does not change the sign of inequality. Also divide by same positive number does not reverse the sign of inequality as shown below

$\begin{array}{c}3p+1-1\le 7-1\\ 3p\le 6\\ \frac{3p}{3}\le \frac{6}{3}\\ p\le 2\end{array}$

- Simplify 2p-9≥7

To simplify given inequality, use addition and division property of inequalities. Add by same number does not change the sign of inequality. Also divide by same positive number does not reverse the sign of inequality as shown below

$\begin{array}{c}2p-9+9\ge 7+9\\ 2p\ge 16\\ \frac{2p}{2}\ge \frac{16}{2}\\ p\ge 8\end{array}$

Thus, solution is $p\le 2\text{or p}\ge \text{8}$

- Graph the solution set

Assolution set of the inequality is all real numbers less than or equal to 2 or greater than or equal to 8

Number line starts from 2 to $-\infty$ and from 8 to $\infty$

Dot on 2 and 8 shows these points are included in the solution set