To solve the inequality 8x - 16x^2 - 24 ≥ 0 correctly, we first need to rewrite the inequality in standard quadratic form with the quadratic term (x^2) coefficient being positive.
8x - 16x^2 - 24 ≥ 0
-16x^2 + 8x - 24 ≥ 0
-16(x^2 - 1/2x + 3/2) ≥ 0
Next, we can factor out a -16 from the quadratic expression inside the parentheses:
-16[(x - 3/4)^2 - 9/16] ≥ 0
Now, we can simplify further:
-16(x - 3/4)^2 + 9/4 ≥ 0
The equation above is the correct form of the inequality. To solve it, we can analyze the sign of the quadratic expression. The quadratic term (x - 3/4)^2 is always non-negative, so the sign of the inequality is determined by the constant term 9/4.
Since the constant term is positive (greater than 0), the inequality holds true for all real values of x. Hence, the solution to the inequality is x belongs to the set of all real numbers, or simply (-∞, ∞).
Therefore, the correct solution to the inequality is x ∈ (-∞, ∞).
Your friend says that the solutions to the inequality 8x - 16x2 - 24 are x≥3. Solve the inequality correctly.
1 answer