Question

solve the compound inequality. use graphs to show the solution set to each of the two given inequalities, as well as a third graph that shows the solution set of the compound inequality. except for the empty set, express the solution set in interval notation.

x - 4 ≤ 2 and 2x + 1 > - 3

graph the solution set of x - 4 ≤ 2. choose the correct graph below.

a. graph with arrow left to a point, b. graph with arrow right from a point, c. graph with arrow right from an open point, d. graph with arrow left to a closed point

graph the solution set of 2x + 1 > - 3. choose the correct graph below.

a. graph with arrow left to a point, b. graph with arrow left to a closed point, c. graph with arrow right from a closed point, d. graph with arrow right from an open point

graph the solution set of the compound inequality x - 4 ≤ 2 and 2x + 1 > - 3. choose the correct graph below.

Explanation:

Step1: Solve \( x - 4 \leq 2 \)

Add 4 to both sides: \( x - 4 + 4 \leq 2 + 4 \)
Simplify: \( x \leq 6 \)
For the graph, a closed dot at 6 (since \( \leq \)) and shading to the left. So the correct graph is D (as D has a closed bracket at 6 and shading left).

Step2: Solve \( 2x + 1 > -3 \)

Subtract 1 from both sides: \( 2x + 1 - 1 > -3 - 1 \)
Simplify: \( 2x > -4 \)
Divide by 2: \( x > -2 \)
For the graph, an open dot at -2 (since \( > \)) and shading to the right. So the correct graph is D (as D has an open bracket at -2 and shading right).

Step3: Solve the compound inequality \( x - 4 \leq 2 \) and \( 2x + 1 > -3 \)

From Step1, \( x \leq 6 \); from Step2, \( x > -2 \). So the solution is \( -2 < x \leq 6 \).
For the graph, open dot at -2, closed dot at 6, shading between them.

Answer:

Graph of \( x - 4 \leq 2 \): D

Graph of \( 2x + 1 > -3 \): D

Graph of compound inequality: (graph with open at -2, closed at 6, shading between)