Question

solve the inequality and graph the solution.
17 ≤ 6(p + 3) - 19
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.
submit

Explanation:

Step1: Simplify the right - hand side

First, expand \(6(p + 3)\) using the distributive property \(a(b + c)=ab+ac\). So \(6(p + 3)-19=6p+18 - 19=6p - 1\).
The inequality becomes \(17\leq6p - 1\).

Step2: Isolate the term with the variable

Add \(1\) to both sides of the inequality. Using the addition property of inequalities \(a\leq b\Rightarrow a + c\leq b + c\).
\(17+1\leq6p-1 + 1\), which simplifies to \(18\leq6p\).

Step3: Solve for \(p\)

Divide both sides of the inequality by \(6\). Using the division property of inequalities \(a\leq b\) and \(c>0\Rightarrow\frac{a}{c}\leq\frac{b}{c}\).
\(\frac{18}{6}\leq\frac{6p}{6}\), so \(3\leq p\) or \(p\geq3\).

Answer:

The solution of the inequality \(17\leq6(p + 3)-19\) is \(p\geq3\). On the number - line, we plot a closed circle at \(p = 3\) (because the inequality is \(\geq\)) and draw a ray to the right of \(3\).