Question

identify the graph of the compound inequality $(-2 < 2x - 6) cap (3x - 12 leq 10)$.

Explanation:

Step1: Solve the left - hand inequality

Solve \(-2<2x - 6\).
Add \(6\) to both sides: \(-2+6<2x-6 + 6\), which simplifies to \(4<2x\).
Divide both sides by \(2\): \(\frac{4}{2}<\frac{2x}{2}\), so \(2 < x\).

Step2: Solve the right - hand inequality

Solve \(2x-6\leqslant3x - 12\).
Subtract \(2x\) from both sides: \(2x-6-2x\leqslant3x - 12-2x\), which gives \(-6\leqslant x - 12\).
Add \(12\) to both sides: \(-6 + 12\leqslant x-12 + 12\), so \(6\leqslant x\).

Combining the two inequalities \(2 < x\) and \(x\geqslant6\), we get \(x\geqslant6\).

Answer:

The graph with a closed circle at \(6\) (since \(x = 6\) is included in the solution set) and an arrow pointing to the right (because \(x\) is greater than or equal to \(6\)) is the correct one.