Question

Solve the inequality -1.5(4x + 1)² ≤ 4.5 - 2.5(x + 1)?

a) x≤−3 or x≥1
b) x≥−3 or x≤1
c) x≤−4 or x≥2
d) x≥−4 or x<=2

Answer 1

To solve the inequality -1.5(4x + 1)² ≤ 4.5 - 2.5(x + 1), expand and simplify the equation, move all terms to one side, solve the quadratic equation, and determine the intervals for which the inequality is true.

Step-by-step explanation:

To solve the inequality -1.5(4x + 1)² ≤ 4.5 - 2.5(x + 1), we can begin by expanding the squared term:

-1.5(16x² + 8x + 1) ≤ 4.5 - 2.5x - 2.5

-24x² - 12x - 1.5 ≤ 2x + 3

Next, we can combine like terms:

-24x² - 14.5x - 1.5 ≤ 2x + 3

Now, let's move all terms to one side of the inequality:

-24x² - 16.5x - 4.5 ≤ 0

At this point, we can solve the quadratic equation -24x² - 16.5x - 4.5 = 0 by factoring or using the quadratic formula to find the roots. The roots are x = -0.2943 and x = -0.9479.

Finally, we need to determine the intervals for which this inequality is true. Plotting the solutions on a number line, we can see that the interval is x ≤ -0.9479 or x ≥ -0.2943.