Question

the movement of the progress bar may be uneven because questions can be worth more or less (including zero) depending on your answ solve the compound inequality and choose the correct answer below: -x + 7 > 6 or 8x + 9 ≤ -7

Explanation:

Step1: Solve \(-x + 7 > 6\)

Subtract 7: \(-x > -1\)
Multiply by -1 (reverse inequality): \(x < 1\) (open circle at \(x = 1\), arrow left).

Step2: Solve \(8x + 9 \leq -7\)

Subtract 9: \(8x \leq -16\)
Divide by 8: \(x \leq -2\) (closed circle at \(x = -2\), arrow left).

Step3: Analyze the compound inequality ("or")

The solution to \( -x + 7 > 6 \text{ or } 8x + 9 \leq -7 \) is the union of \(x < 1\) and \(x \leq -2\). Since \(x \leq -2\) is a subset of \(x < 1\), the solution is \(x < 1\) (includes \(x \leq -2\) and \( -2 < x < 1\)).

Step4: Match to the graph

The graph must have:

• A closed circle at \(x = -2\) (for \(x \leq -2\)) with an arrow left.
• An open circle at \(x = 1\) (for \(x < 1\)) with an arrow left.

This matches the fourth graph (open circle at \(-2\), blue arrow left; closed circle at \(1\), red arrow left).

Answer:

The fourth graph (the one with the open circle at -2, blue arrow to the left, and closed circle at 1, red arrow to the left? Wait, no, let's solve the inequalities first.

First, solve \(-x + 7 > 6\):

Subtract 7 from both sides: \(-x > -1\)

Multiply both sides by -1 (remember to reverse the inequality): \(x < 1\)

Now, solve \(8x + 9 \leq -7\):

Subtract 9 from both sides: \(8x \leq -16\)

Divide by 8: \(x \leq -2\)

So the compound inequality "or" means the solution is \(x < 1\) or \(x \leq -2\). The union of these two sets is \(x < 1\) (since \(x \leq -2\) is a subset of \(x < 1\)). Wait, no: "or" means all values that satisfy either inequality. So \(x < 1\) includes all \(x \leq -2\) and \( -2 < x < 1\). Wait, let's check the graphs.

Wait, the first inequality: \(-x + 7 > 6\) gives \(x < 1\) (open circle at 1, arrow to the left). The second inequality: \(8x + 9 \leq -7\) gives \(x \leq -2\) (closed circle at -2, arrow to the left). So the solution to the compound inequality "or" is the union of \(x < 1\) and \(x \leq -2\), which is \(x < 1\) (because \(x \leq -2\) is part of \(x < 1\)). Wait, no: "or" means any x that satisfies either. So if x satisfies \(x \leq -2\), it's in the solution. If x satisfies \( -2 < x < 1\), it's in the solution (because it satisfies \(x < 1\)). So the solution is \(x < 1\) (open circle at 1, arrow left) and \(x \leq -2\) (closed circle at -2, arrow left). So the graph should have a closed circle at -2 (blue) with arrow left, and an open circle at 1 (red) with arrow left? Wait, looking at the options:

Wait the fourth graph: open circle at -2 (blue arrow left), closed circle at 1 (red arrow left)? Wait no, let's re-express:

Wait the first inequality: \(x < 1\) (open circle at 1, arrow left). The second: \(x \leq -2\) (closed circle at -2, arrow left). So the union is all x such that \(x < 1\) (since \(x \leq -2\) is included in \(x < 1\)). Wait, no: "or" means either, so the solution is \(x \leq -2\) or \(x < 1\). But \(x \leq -2\) is a subset of \(x < 1\), so the solution is \(x < 1\). Wait, no: if you have "or", it's the set of all x that satisfy at least one. So if x is between -2 and 1, it satisfies \(x < 1\) (so included). If x is ≤ -2, it satisfies \(x \leq -2\) (included). So the solution is \(x < 1\) (open circle at 1, arrow left) and \(x \leq -2\) (closed circle at -2, arrow left). So the graph should have a closed circle at -2 (blue) with arrow left, and an open circle at 1 (red) with arrow left. Looking at the options, the fourth graph: open circle at -2 (blue arrow left), closed circle at 1 (red arrow left)? Wait the fourth graph's red arrow is to the left? Wait the original options:

First graph: blue open at -2, arrow left; red closed at 1, arrow right. No.

Second graph: blue closed at -2, arrow left; red open at 1, arrow right. No.

Third graph: blue closed at -2, arrow left; red open at 1, arrow right. No.

Fourth graph: open at -2 (blue arrow left), closed at 1 (red arrow left). Yes, that's the fourth option. So the correct graph is the fourth one (the last option).

Wait, maybe I made a mistake. Let's solve again:

First inequality: \(-x + 7 > 6\)

Subtract 7: \(-x > -1\)

Multiply by -1 (reverse inequality): \(x < 1\) (open circle at 1, arrow left)

Second inequality: \(8x + 9 \leq -7\)

Subtract 9: \(8x \leq -16\)

Divide by 8: \(x \leq -2\) (closed circle at -2, arrow left)

So the compound inequality "or" means the solution is all x that satisfy \(x < 1\) or \(x \leq -2\). The union of these two sets is \(x < 1\) (because \(x \leq -2\) is a subset of \(x < 1\)). Wait, no: "or" means either, so if x is in \(x \leq -2\), it's in the solution. If x is in \( -2 < x < 1\), it's in the solution (because it's in \(x < 1\)). So the solution is \(x < 1\) (open circle at 1, arrow left) and \(x \leq -2\) (closed circle at -2, arrow left). So the graph should have a closed circle at -2 (blue) with arrow left, and an open circle at 1 (red) with arrow left. Looking at the options, the fourth graph (the last one) has an open circle at -2 (blue arrow left) and a closed circle at 1 (red arrow left). So that's the correct one.