Question

for questions 23-28, select the inequality or compound inequality that describes the graph
23.
a. ( x leq -9 ) or ( x > -3 )
b. ( -9 leq x > -3 )
c. ( x leq -9 ) or ( x < -3 )
d. ( x geq -9 ) or ( x < -3 )
24.
a. ( x > -1 )
b. ( x geq -1 )
c. ( x leq -1 )
d. ( x < -1 )

Explanation:

Question 23

Step1: Analyze the graph's direction and endpoint

The graph has a leftward arrow (including -10, -15, etc.) and a closed dot at -10? Wait, no, looking at the number line: the first graph (Q23) has a segment with a closed dot (so inclusive) at -10? Wait, the options have -9 and -3. Wait, maybe typo, but let's check the options. The graph for Q23: the left part is a ray ending at -10? Wait, the options have -9. Maybe the number line is marked with -25, -20, -15, -10, -5, 0. Wait, maybe the endpoint is -10, but options have -9. Maybe a typo, but let's check the inequality direction. The left ray (including the endpoint) and then another? Wait, no, the graph for Q23: the left part is a ray going left from -10 (closed dot) and the right part? Wait, the options are compound inequalities. Let's check each option:

Option A: \( x \leq -9 \) or \( x > -3 \) – but the graph's left part is up to -10 (closed), maybe -9. Wait, maybe the endpoint is -9. Let's see the direction: the left ray (including -9, so \( x \leq -9 \)) and the right ray? No, the graph shows a left ray (closed dot) and then a right ray? Wait, no, the first graph (Q23) has a leftward arrow (including the endpoint) and then a rightward? Wait, the options:

Option B: \( -9 \leq x > -3 \) – invalid, compound inequality should be "and" or "or", but this is mixed.

Option C: \( x \leq -9 \) or \( x < -3 \) – no, the right part should be open?

Wait, maybe the graph for Q23: the left ray is \( x \leq -9 \) (closed dot) and the right ray is \( x > -3 \) (open dot)? Wait, the options:

Wait, the correct approach: on a number line, a closed dot means inclusive (\( \leq \) or \( \geq \)), open dot means exclusive (\( < \) or \( > \)). The first graph (Q23) has a closed dot at -9 (so \( x \leq -9 \)) and an open dot at -3 (so \( x > -3 \))? Wait, option A is \( x \leq -9 \) or \( x > -3 \). Let's check the graph: the left part is a ray with closed dot (inclusive) at -9, going left, and the right part is a ray with open dot at -3, going right. So that matches option A.

Step2: Eliminate other options

Option B: \( -9 \leq x > -3 \) is invalid (can't have \( \leq \) and \( > \) in the same variable). Option C: \( x \leq -9 \) or \( x < -3 \) – the right part should be \( > -3 \) (open dot, so \( x > -3 \)). Option D: \( x \geq -9 \) or \( x < -3 \) – left part is \( \geq \), but graph is \( \leq \). So correct is A.

Step3: Confirm

So Q23 answer is A.

Question 24

Step1: Analyze the graph's direction and endpoint

The graph for Q24 has a leftward arrow with a closed dot at -1. So the inequality is \( x \leq -1 \) (since the dot is closed, inclusive, and the arrow is left, so values less than or equal to -1).

Step2: Check options

Option A: \( x > -1 \) – arrow right, open dot, no.

Option B: \( x \geq -1 \) – arrow right, closed dot, no.

Option C: \( x \leq -1 \) – closed dot at -1, arrow left, correct.

Option D: \( x < -1 \) – open dot, but graph has closed dot, so no.

Answer:

1. A. \( x \leq -9 \) or \( x > -3 \)
2. C. \( x \leq -1 \)