Solve the inequalities -6 $\leq$ 4 - 2x < 5 - x

Analyze the question: This is a compound inequality: -6 ≤ 4 - 2x < 5 - x. We split it into two inequalities, solve each, and then find where both are true.

Concept: When you multiply or divide an inequality by a negative number, the inequality sign flips.

Solution:

Part 1: -6 ≤ 4 - 2x

• Subtract 4 from both sides: -10 ≤ -2x
• Divide by -2 (flip the sign): 5 ≥ x, so x ≤ 5

Part 2: 4 - 2x < 5 - x

• Add 2x to both sides: 4 < 5 + x
• Subtract 5: -1 < x, so x > -1

Combine: x > -1 and x ≤ 5, so

-1 < x ≤ 5

Check:

• x = 5: 4 - 10 = -6, and -6 ≤ -6 is true. Also -6 < 0 is true. So x = 5 is included.
• x = -1: 4 + 2 = 6 and 5 + 1 = 6, so 6 < 6 is false. So x = -1 is excluded.
• x = 0: -6 ≤ 4 < 5 is true.

Tip: Treat a compound inequality as two separate inequalities. Remember to flip the sign when dividing by a negative. Then test the endpoints to decide whether each is included (≤) or excluded (<).