Question

solve the inequality and graph the solution.
8(p - 1) - 10 > 6
plot the endpoints. select an endpoint to change it from closed to open. select the middle of the segment, ray, or line to delete it.

Explanation:

Step1: Simplify the left - hand side

First, expand \(8(p - 1)\) using the distributive property \(a(b - c)=ab - ac\). So \(8(p - 1)-10=8p-8 - 10=8p-(8 + 10)=8p-18\).
The inequality becomes \(8p-18>6\).

Step2: Isolate the term with \(p\)

Add \(18\) to both sides of the inequality. Using the addition property of inequalities \(a>b\Rightarrow a + c>b + c\).
\(8p-18 + 18>6+18\), which simplifies to \(8p>24\).

Step3: Solve for \(p\)

Divide both sides of the inequality by \(8\). Using the division property of inequalities \(a>b,c>0\Rightarrow\frac{a}{c}>\frac{b}{c}\).
\(\frac{8p}{8}>\frac{24}{8}\), so \(p > 3\).

To graph the solution:

• The endpoint is \(p = 3\). Since the inequality is \(p>3\) (not \(p\geq3\)), the endpoint at \(p = 3\) is open.
• The solution is the ray starting at \(p = 3\) and extending to the right (towards larger values of \(p\)).

Answer:

The solution of the inequality \(8(p - 1)-10>6\) is \(p>3\). On the number - line, there is an open circle at \(3\) and a ray extending to the right.