Question

1. cheyenne graphs the inequality $2x + 4 \leq 8$ or $x + 8 \geq 12$. what is her graph?

(options with number lines: first has solid dots at 2 and 4, second solid at 2 and open at 4, third open at both, fourth open at 2 and solid at 4)

Explanation:

Step1: Solve \(2x + 4 \leq 8\)

Subtract 4 from both sides: \(2x \leq 8 - 4\) → \(2x \leq 4\). Divide by 2: \(x \leq 2\).

Step2: Solve \(x + 8 \geq 12\)

Subtract 8 from both sides: \(x \geq 12 - 8\) → \(x \geq 4\).

Step3: Analyze the graphs

The solution is \(x \leq 2\) or \(x \geq 4\). So we need a graph with a closed dot at 2 (since \(x \leq 2\)) and a closed dot at 4 (since \(x \geq 4\)), with lines extending left from 2 and right from 4. The first graph has closed dots at 2 and 4, with the line between? Wait, no—wait, "or" means the union. Wait, no: \(x \leq 2\) is left of 2, \(x \geq 4\) is right of 4. Wait, the first graph shows a line from 2 to 4, but that's \(2 \leq x \leq 4\), which is "and". Wait, no, I must have messed up. Wait, no: \(2x + 4 \leq 8\) is \(x \leq 2\), \(x + 8 \geq 12\) is \(x \geq 4\). So the solution is \(x \leq 2\) or \(x \geq 4\). So the graph should have a closed dot at 2 (since \(x \leq 2\) includes 2) and a closed dot at 4 (since \(x \geq 4\) includes 4), with the line to the left of 2 and to the right of 4. Wait, the first option's graph: let's check the dots. First graph: closed at 2 and 4, line from 2 to 4? No, that's \(2 \leq x \leq 4\), which is "and". Wait, no, maybe I made a mistake in solving. Wait, \(2x + 4 \leq 8\): subtract 4: \(2x \leq 4\), divide by 2: \(x \leq 2\). Correct. \(x + 8 \geq 12\): subtract 8: \(x \geq 4\). Correct. So "or" means the union of \(x \leq 2\) and \(x \geq 4\). So the graph should have a closed circle at 2 (because \(x \leq 2\) includes 2) and a closed circle at 4 (because \(x \geq 4\) includes 4), with the line extending to the left from 2 and to the right from 4. Wait, the first graph: let's look at the options. First graph: numbers -4, -2, 0, 2, 4, 6, 8, 10. The blue line is from -4 (left) up to 2, then from 2 to 4, then from 4 to 10? No, the first graph's blue line: left arrow at -4, then a line from -4 to 2 (closed dot), then from 2 to 4 (closed dot), then from 4 to 10. Wait, that would be \(x \leq 4\)? No, maybe I misread. Wait, the first graph: the blue line is a single line from -4 to 10, with closed dots at 2 and 4. Wait, no, the first graph's blue line: let's see, the first option's number line: the blue line starts at the left arrow (so all left of 2) and then from 2 to 4, then from 4 to the right? No, that can't be. Wait, maybe the first graph is actually showing \(x \leq 2\) or \(x \geq 4\) with closed dots. Wait, maybe the first graph has a closed dot at 2 (so \(x \leq 2\)) and a closed dot at 4 (so \(x \geq 4\)), with the line between them? No, that's \(2 \leq x \leq 4\), which is "and". Wait, I'm confused. Wait, the problem says "or", so it's the union. So the solution is all numbers less than or equal to 2, or greater than or equal to 4. So on the number line, we have a closed circle at 2 (since \(x \leq 2\) includes 2) and a closed circle at 4 (since \(x \geq 4\) includes 4), with the line extending to the left from 2 (for \(x \leq 2\)) and to the right from 4 (for \(x \geq 4\)). The first graph: let's check the dots. First graph: closed at 2 and 4, with the blue line covering from -4 to 2 (left of 2) and from 4 to 10 (right of 4), and the line between 2 and 4? No, that would be a solid line from 2 to 4, which is \(2 \leq x \leq 4\), but that's "and". Wait, maybe the first graph is actually the correct one because the "or" here is misinterpreted? Wait, no—wait, maybe I made a mistake in the inequality direction. Wait, \(2x + 4 \leq 8\): \(2x \leq 4\) → \(x \leq 2\). Correct. \(x + 8 \geq 12\) → \(x \geq 4\). Correct. So "or" means either \(x \leq 2\) or \(x \geq 4\). So the graph should have a closed dot at 2 (since \(x \leq 2\)) and a closed dot at 4 (since \(x \geq 4\)), with the line to the left of 2 and to the right of 4. The first graph has closed dots at 2 and 4, and the blue line is from -4 (left) to 2, then from 2 to 4, then from 4 to 10. Wait, that would be all real numbers, but that's not right. Wait, no—maybe the first graph is the only one with closed dots at 2 and 4, which matches the inequalities (since both inequalities are "less than or equal" and "greater than or equal", so closed dots). The other graphs have open dots, which would be for strict inequalities. So the first graph is the one with closed dots at 2 and 4, so that's the correct one.

Answer:

The first graph (the topmost one with closed dots at 2 and 4)