Question

Solve the inequalities: (x−1)(x+2)(2x−7)²≤0

Answer 1

To solve the inequality (x−1)(x+2)(2x−7)²≤0, first find the critical numbers and test values from each interval to determine the solution.

Step-by-step explanation:

To solve the inequality (x−1)(x+2)(2x−7)²≤0, we first need to find the critical numbers, which are the values of x that make each factor equal to zero.

1. Setting x-1=0, we find x=1.
2. Setting x+2=0, we find x=-2.
3. Setting 2x-7=0, we find x=7/2.

These critical numbers divide the number line into four intervals: (-∞,-2), (-2,1), (1,7/2), and (7/2,∞). We then test a value from each interval to see whether the inequality is true or false. By doing so, we find that the solution to the inequality is x∈(-∞,-2]∪(1,7/2].