Question

which graph represents the compound inequality below?
$7x + 2 \leq 16$ or $-3x - 10 \leq -34$
show your work here
hint: to add the intersection symbol ($\cap$), type \intersection\
(there are four number lines with different markings as options, but the specific details of the number lines are as follows: first number line: from -10 to 10, with a blue dot at some point and a blue open circle at another, line connecting them; second number line: from -10 to 10, blue line pointing left with a blue dot; third number line: from -10 to 10, blue line pointing left and right with two blue dots; fourth number line: from -10 to 10, blue line pointing right with a blue dot)

Explanation:

Step1: Solve \(7x + 2\leq16\)

Subtract \(2\) from both sides: \(7x\leq16 - 2\), so \(7x\leq14\).
Divide both sides by \(7\): \(x\leq2\).

Step2: Solve \(-3x - 10\leq - 34\)

Add \(10\) to both sides: \(-3x\leq-34 + 10\), so \(-3x\leq-24\).
Divide both sides by \(-3\) (and reverse the inequality sign): \(x\geq8\).

The compound inequality \(7x + 2\leq16\) or \(-3x - 10\leq - 34\) means \(x\leq2\) or \(x\geq8\).

Answer:

The graph that has a solid dot at \(x = 2\) with a line extending to the left (for \(x\leq2\)) and a solid dot at \(x = 8\) with a line extending to the right (for \(x\geq8\)) is the correct one. (Assuming the third option in the original multiple - choice setup, as per the described solution \(x\leq2\) or \(x\geq8\))