Question

Solve each inequality. Then graph it on a number line.

$\mathbf{13}\mathbf{-}\mathit{p}\mathbf{\ge }\mathbf{15}$

Short answer

The solution of the given inequality$13-p\ge 15$ is $\mathit{p}\mathbf{\le }\mathbf{-}\mathbf{2}$.

The graph of the solution of the given inequality$13-p\ge 15$ that is graph of$p\le -2$ is:

Step-by-step solution

Step 1. Write the subtraction property and multiplication property of inequalities.

The subtraction property of inequalities states that if the same number is subtracted from each side of a true inequality, the resulting inequality is also true that is:

(i) If $a>b$, then $a-c>b-c$.

(ii) If $a<b$, then $a-c<b-c$.

The multiplication property of inequalities states that if both sides of the inequality are multiplied by a positive number the sign of the inequality remains the same and if both sides of the inequality are multiplied by a negative number then the sign of the inequality changes that is:

(i) If $a>b$and role="math" localid="1647496591700" $c$is a positive number then $ac>bc$.

(ii) If $a<b$and$c$is a positive number then $ac<bc$.

(ii) If $a>b$and$c$ is a negative number then $ac<bc$.

(iv) If$a<b$ and$c$ is a negative number then $ac>bc$.

Solve the given inequality 13−p≥15.

The solution of the given inequality$13-p\ge 15$ is:

$\begin{array}{l}13-p\ge 15\\ 13-p-13\ge 15-13\left(\text{byusingsubtractionpropertyofinequality}\right)\\ -p+13-13\ge 2\\ -p\ge 2\\ (-1)(-p)\le 2(-1)\left(\text{byusingmutiplicationpropertyofinequality}\right)\\ p\le -2\end{array}$

Therefore, the solution of the given inequality$13-p\ge 15$ is $p\le -2$.

Step 3. Draw the graph of the inequality p≤−2.

The graph of the solution of the given inequality $13-p\ge 15$that is graph of $p\le -2$is:

It can be noticed that at $p=-2$, there is closed circle because$-2$ is included in the inequality $p\le -2$.