Question

identify the graph of the compound inequality (-2 < 2x - 6) ∩ (3x - 12 ≤ 10).

Explanation:

Step1: Solve the first 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 second inequality

Solve $3x-12\leqslant10$.
Add 12 to both sides: $3x-12 + 12\leqslant10 + 12$, which gives $3x\leqslant22$.
Divide both sides by 3: $x\leqslant\frac{22}{3}\approx7.33$.

Step3: Combine the solutions

The compound - inequality $(-2 < 2x - 6)\cap(3x - 12\leqslant10)$ has the solution $2 < x\leqslant\frac{22}{3}$.
On a number - line, we have an open circle at $x = 2$ (because $x>2$) and a closed circle at $x=\frac{22}{3}\approx7.33$ and the shaded region in between.

Answer:

The graph with an open circle at 2 and a closed circle at approximately 7.33 (around 7.33 on the number - line) and the region between them shaded.