Question

solve the inequality. graph the solution set.
2 ≤ 4x - 14 ≤ 0
choose the correct graph below.

Explanation:

Step1: Split the compound - inequality

We have the compound inequality \(2\leq4x - 14\leq0\). First, split it into two inequalities: \(2\leq4x - 14\) and \(4x - 14\leq0\).

Step2: Solve \(2\leq4x - 14\)

Add 14 to both sides: \(2+14\leq4x-14 + 14\), which simplifies to \(16\leq4x\). Then divide both sides by 4: \(\frac{16}{4}\leq\frac{4x}{4}\), so \(4\leq x\).

Step3: Solve \(4x - 14\leq0\)

Add 14 to both sides: \(4x-14 + 14\leq0 + 14\), which gives \(4x\leq14\). Then divide both sides by 4: \(x\leq\frac{14}{4}=\frac{7}{2}=3.5\).
However, there is a contradiction since \(x\) cannot be both \(\geq4\) and \(\leq3.5\) simultaneously. So the solution - set is the empty set \(\varnothing\).
Since there is no graph for the empty - set among the options provided (assuming the graphs represent non - empty intervals), there is an error in the problem setup as written. If we assume the compound inequality was \(2\leq4x - 14\leq10\):

Step1: Split the compound - inequality

Split \(2\leq4x - 14\leq10\) into \(2\leq4x - 14\) and \(4x - 14\leq10\).

Step2: Solve \(2\leq4x - 14\)

Add 14 to both sides: \(2 + 14\leq4x-14+14\), so \(16\leq4x\). Divide both sides by 4: \(x\geq4\).

Step3: Solve \(4x - 14\leq10\)

Add 14 to both sides: \(4x-14 + 14\leq10 + 14\), so \(4x\leq24\). Divide both sides by 4: \(x\leq6\).
The solution set is \(4\leq x\leq6\).

Answer:

If the correct inequality is \(4\leq x\leq6\), the graph would be a line segment on the number line starting at 4 (closed - circle) and ending at 6 (closed - circle). Without seeing the actual graphs with proper labels, we can't choose from the given options A, B, C, D precisely, but the solution set is \([4,6]\) on the number line. If the original inequality \(2\leq4x - 14\leq0\) is correct, the solution set is \(\varnothing\).