Question

identifying inequalities with no solution
identify the graph of the compound inequality $(-2 < 2x - 10) cap (3x - 12 leq 10)$

Explanation:

Step1: Solve the first inequality

Solve \(-2 < 2x - 10\).
Add \(10\) to both sides: \(-2+10<2x - 10 + 10\), which simplifies to \(8 < 2x\).
Divide both sides by \(2\): \(\frac{8}{2}<\frac{2x}{2}\), so \(4 < x\) or \(x>4\).

Step2: Solve the second inequality

Solve \(3x - 12\leq10\).
Add \(12\) to both sides: \(3x-12 + 12\leq10 + 12\), which simplifies to \(3x\leq22\).
Divide both sides by \(3\): \(x\leq\frac{22}{3}\approx7.33\).

Answer:

The solution of the compound inequality \(-2 < 2x - 10\) and \(3x - 12\leq10\) is \(4 < x\leq\frac{22}{3}\). So, we look for a graph where there is an open - circle at \(x = 4\) (because \(x>4\)) and a closed - circle at \(x=\frac{22}{3}\approx7.33\) (because \(x\leq\frac{22}{3}\)) with the line segment connecting these two points.