Question

Given the inequality |−2x + 5| + 6 ≤ 4, you want to solve for x. You're considering multiplying both sides by -1 to eliminate the negative number on the right and potentially obtain two solutions. However, the teacher's edition shows no solution, the school teacher says all real numbers are solutions, and some apps show two solutions. Which approach is the correct procedure? Explain your reasoning.

Answer 1

To solve the inequality |−2x + 5| + 6 ≤ 4, you should approach it algebraically and avoid multiplying both sides by -1. The correct procedure involves isolating the absolute value, splitting the inequality into cases, solving for x in each case, and combining the solutions.

Step-by-step explanation:

The correct procedure to solve the inequality |−2x + 5| + 6 ≤ 4 is to approach it algebraically without multiplying both sides by -1. Here's the step-by-step explanation:

1. Start by subtracting 6 from both sides to isolate the absolute value: |−2x + 5| ≤ -2.
2. Next, split the inequality into two cases: -2x + 5 ≤ -2 and -(-2x + 5) ≤ -2.
3. For the first case, solve for x: -2x + 5 ≤ -2. Subtract 5 from both sides and divide by -2 to get x ≥ 3.5.
4. For the second case, solve for x: 2x - 5 ≤ -2. Add 5 to both sides and divide by 2 to get x ≤ 1.5.
5. Combining the two cases, the inequality is satisfied when x ≤ 1.5 or x ≥ 3.5.