Question

Solve the inequality: x + 7 < 3x + 3 ≤ x + 2. If there is no solution for the inequality, please choose the correct option.

A) x < -2
B) -2 < x < 2
C) x > 2
D) No Solution

Answer 1

To solve the compound inequality x + 7 < 3x + 3 ≤ x + 2, we split it into two separate inequalities and solve each one. However, these inequalities yield non-overlapping solutions, leading to the conclusion that there is no value of x that satisfies both, and thus there is no solution to the combined inequality. Therefore, the correct option is D) No Solution.

Step-by-step explanation:

To solve the inequality x + 7 < 3x + 3 ≤ x + 2, we need to treat it as two separate inequalities:

• x + 7 < 3x + 3
• 3x + 3 ≤ x + 2

Let's solve each inequality step-by-step:

1. For x + 7 < 3x + 3, subtract x from both sides to get 7 < 2x + 3.
2. Subtract 3 from both sides to get 4 < 2x.
3. Divide both sides by 2 to obtain 2 < x, or x > 2.

Now let's solve the second inequality:

1. For 3x + 3 ≤ x + 2, subtract x from both sides to get 2x + 3 ≤ 2.
2. Subtract 3 from both sides to get 2x ≤ -1.
3. Divide both sides by 2 to obtain x ≤ -0.5.

Comparing the solutions of both inequalities, we see that there is no overlap. The first inequality is only true when x > 2, while the second is only true when x ≤ -0.5. Since there is no value of x that satisfies both conditions, there is No Solution to the combined inequality. Therefore, the correct option is D) No Solution.