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select all the inequalities for which the solutions are shown on the graph.
a) $-1 + 6t \leq 4(t + 3) + 3$
b) $-2(t + 11) - 3t - 21 \leq 3(t + 7)$
c) $-5(t + 4) + 6t + 18 > -10$
d) $6(t - 1) - 19 \geq 23$
e) $3t - 10 \leq 4t - 6 + 4$

Explanation:

First, we determine the solution from the graph. The graph has a closed dot at -8 and the line goes to the right, so the solution is \( t \geq -8 \). Now we solve each inequality:

Step 1: Solve Option A

Inequality: \( -1 + 6t \leq 4(t + 3) + 3 \)
Expand right side: \( -1 + 6t \leq 4t + 12 + 3 \)
Simplify: \( -1 + 6t \leq 4t + 15 \)
Subtract \( 4t \): \( -1 + 2t \leq 15 \)
Add 1: \( 2t \leq 16 \)
Divide by 2: \( t \leq 8 \). Not \( t \geq -8 \).

Step 2: Solve Option B

Inequality: \( -2(t + 11) - 3t - 21 \leq 3(t + 7) \)
Expand: \( -2t - 22 - 3t - 21 \leq 3t + 21 \)
Simplify left: \( -5t - 43 \leq 3t + 21 \)
Add \( 5t \): \( -43 \leq 8t + 21 \)
Subtract 21: \( -64 \leq 8t \)
Divide by 8: \( -8 \leq t \) or \( t \geq -8 \). Matches the graph.

Step 3: Solve Option C

Inequality: \( -5(t + 4) + 6t + 18 > -10 \)
Expand: \( -5t - 20 + 6t + 18 > -10 \)
Simplify: \( t - 2 > -10 \)
Add 2: \( t > -8 \). The graph has a closed dot at -8, so this is not a match (since it's \( t > -8 \), not \( t \geq -8 \)).

Step 4: Solve Option D

Inequality: \( 6(t - 1) - 19 \geq 23 \)
Expand: \( 6t - 6 - 19 \geq 23 \)
Simplify: \( 6t - 25 \geq 23 \)
Add 25: \( 6t \geq 48 \)
Divide by 6: \( t \geq 8 \). Not \( t \geq -8 \).

Step 5: Solve Option E

Inequality: \( 3t - 10 \leq 4t - 6 + 4 \)
Simplify right: \( 3t - 10 \leq 4t - 2 \)
Subtract \( 3t \): \( -10 \leq t - 2 \)
Add 2: \( -8 \leq t \) or \( t \geq -8 \). Matches the graph.

Answer:

B. \( -2(t + 11) - 3t - 21 \leq 3(t + 7) \), E. \( 3t - 10 \leq 4t - 6 + 4 \)