Question

Solve the inequality and graph the solution. Then check graphically whether the given \(x\) -value is a solution by graphing the \(x\) -value on the same number line. $$ -1<7 x-15 \leq 20 ; x=5 $$

Short answer

The solution to the inequality is \(2 < x \leq 5\) and the given \(x = 5\) is a solution.

Step-by-step solution

Solve the Inequality

We have two inequalities to solve: \(7x-15> -1\) and \(7x-15 \leq 20\). For the first inequality \(7x-15> -1\), we add 15 to both sides to isolate the term with \(x\) on one side. This gives \(7x > 14\), then we divide both sides by 7 to solve for \(x\), resulting in \(x > 2\). Similarly, for \(7x-15 \leq 20\), we add 15 to both side to give \(7x \leq 35\). Dividing by 7 leads to \(x \leq 5\). Combining these, the solution of the inequality is \(2 < x \leq 5\).

Graph the Solution

Plot a number line and mark the points at \(x = 2\) and \(x = 5\). The solution does not include \(x = 2\), so mark with an open circle and for \(x = 5\), mark with a filled circle, as the solution includes \(x = 5\). The solution will be the part of the number line between these two points.

Check the Given \(x = 5\)

The provided \(x = 5\) falls on the filled circle (the solution includes \(x = 5\)) so it is a solution of the given inequality. Checking it in the inequality, for \(x = 5\), \(7x - 15 = 20\) is true, thus it satisfies the inequality.