Solve each inequality algebraically:?

Question

a) x^2 + 5x + 6 < (less than or equal) 0

b) 3/ x^2-4 > 0

Answer 1

a).

The inequality is  .

.

Solution set of the inequality is { x | x : x  ≤ - 3, - 2 }.

b).

The inequality is 3 / x ^2 - 4 > 0.

3 / x ^2 > 4

4x ^2 < 3

x ^2 < 3 / 4

x < ±( √3 / 2 ).

Solution set of the inequality is { x | x : x < ±√3/2}.

Answer 2

If the inequality ax² + bx + c ≤ 0 then the solution set is { x | x₁ ≤ x ≤ x₂ }.

If the inequality ax² + bx + c > 0 then the solution set is { x | x < x₁ or x > x₂ }.

(a).

The inequality is x² + 5x + 6 ≤ 0.

First solve the related quadratic equation x² + 5x + 6 = 0.

x² + 2x + 3x + 6 = 0

x(x + 2) + 3(x + 2) = 0

(x + 3)(x + 2) = 0

x + 3 = 0 or x + 2 = 0

x = - 3 or x = - 2.

For ≤, include the x - intercepts in the solution.

The solution set is { x | - 3 ≤ x ≤ - 2 }.

(b).

The inequality is 3/x² - 4 > 0.

Let the finction f(x) = 3/x² - 4, where x ≠ 0.

Write the inequality 3/x² - 4 > 0 in the form of ax² + bx + c < 0 or ax² + bx + c > 0.

3/x² > 4

3 > 4x²

0 > 4x² - 3

4x² - 3 < 0.

Solve the related quadratic equation 4x² - 3 = 0.

4x² = 3

x² = 3/4

x = ± √3/2.

For <, exclude the x - intercepts in the solution.

The solution set is { x | - √3/2 < x < √3/2, x ≠ 0 }.