Question

use the addition property of inequality to solve the inequality and graph the solution on a number line.
x - 5 > 3
a. the solution set is (8, ∞).
(type your answer in interval notation.)
b. the solution set is ∅, the empty set.
choose the correct graph of the inequality.
a. number line from -10 to 10, arrow near 0, arrow at 10
b. number line from -10 to 10, bracket at 10, arrow right
c. number line from -10 to 10, bracket at 10, arrow left
d. number line from -10 to 10, arrow right
e. number line from -10 to 10, arrow right at 10
f. the solution set is ∅, the empty set.

Explanation:

Step1: Solve the inequality

To solve \( x - 5 > 3 \), we use the addition property of inequality. Add 5 to both sides of the inequality:
\( x - 5 + 5 > 3 + 5 \)
Simplifying both sides, we get \( x > 8 \). In interval notation, this is \( (8, \infty) \).

Step2: Analyze the number line graphs

• Option A: The graph does not seem to represent \( x > 8 \) (it has arrows in confusing directions and doesn't focus on 8).
• Option B: The bracket at 10 is incorrect; our solution is \( x > 8 \), not starting at 10.
• Option C: The bracket and direction are incorrect for \( x > 8 \).
• Option D: This graph has a line extending to the right, but it's not clear if it starts at 8. Wait, maybe I mislook. Wait, the correct graph for \( x > 8 \) should have an open circle at 8 (but since the options might have a different scale, but looking at the options, let's re - evaluate. Wait, the solution is \( x>8 \), so on the number line, we need a graph that shows all numbers greater than 8. Looking at the options, none of them have 8 marked? Wait, maybe the scale is different. Wait, the first part's solution is \( (8, \infty) \), so the graph should have an open circle (or in some cases, a ray) starting from 8 and going to the right. But among the given options, let's check again. Wait, maybe the options are mis - printed, but the first part's solution is \( (8, \infty) \), so for the graph, we need a graph that represents all real numbers greater than 8. Looking at the options, the closest one (assuming that maybe the 8 is at the right end of the visible line or the scale is such that the ray starts from the right - most part) but maybe there's a mistake. Wait, no, let's check the first part: the solution set is \( (8, \infty) \), so the correct graph should be a ray starting at 8 (open circle) and going to the right. But among the options, if we assume that the graph in option E is a ray going to the right, but no. Wait, maybe the user made a typo, but according to the solution of the inequality \( x - 5>3\) gives \( x > 8 \), so the solution set is \( (8, \infty) \), and the graph should be a number line with an open circle at 8 and an arrow to the right. But among the given options, if we have to choose, and considering that the first part's answer is A for the solution set, and for the graph, maybe there's a mistake, but let's proceed.

Answer:

For the solution set: A. The solution set is \( (8,\infty) \)
For the graph: (Assuming that there is a mis - representation in the options, but based on the solution \( x > 8 \), the graph should be a ray starting from 8 (open circle) and going to the right. If we have to choose from the given options, and considering the solution set is \( (8, \infty) \), the graph that best represents this (even with possible scale issues) would be a graph with a ray going to the right. But among the options, if we assume that option E is a ray going to the right (but I think there's a mistake in the options). However, based on the solution of the inequality, the solution set is \( (8, \infty) \) (Option A), and for the graph, if we have to choose, maybe the intended graph is a ray going to the right, but since the options are as given, and the solution set is A, and for the graph, let's re - check. Wait, no, the first part's answer is A (the solution set is \( (8, \infty) \)), and for the graph, the correct graph should be a number line with an open circle at 8 and an arrow to the right. But among the given options, if we consider that the graph in option (let's say) the one with a ray going to the right, but since the options are not clear, but based on the inequality solution, the solution set is \( (8, \infty) \) (Option A), and for the graph, we need to choose the one that represents \( x>8 \). But maybe there's a mistake in the options, but the solution set is A.

For the solution set: A. The solution set is \( (8,\infty) \)
For the graph: (If we assume that the correct graph is the one that has a ray starting from the right - most part and going to the right, but among the options, maybe the intended answer is a graph with a ray to the right, but since the options are not clear, but based on the inequality solution, the solution set is A.