factor 2𝑥^2 − 5𝑥 − 12 ≥ 0
To factorize the given expression, we need to find two numbers whose product is -24 (the coefficient of x^2 multiplied by the constant term) and whose sum is -5 (the coefficient of x). In this case, the numbers are -8 and 3. Thus, we can rewrite the expression as:
2𝑥^2 − 5𝑥 − 12 = (2𝑥 + 3)(𝑥 − 4)
Now, we need to determine when this expression is greater than or equal to zero. To do this, we can analyze the signs of the factors:
(2𝑥 + 3) ≥ 0
𝑥 − 4 ≥ 0
For the first factor, 2𝑥 + 3 ≥ 0, we solve for x:
2𝑥 + 3 ≥ 0
2𝑥 ≥ -3
𝑥 ≥ -3/2
For the second factor, 𝑥 − 4 ≥ 0, we solve for x:
𝑥 − 4 ≥ 0
𝑥 ≥ 4
To determine the values of x that satisfy both conditions (i.e., when both factors are greater than or equal to zero), we can create an interval diagram:
-----------o----------o----------
-3/2 4
From the diagram, we see that the expression is greater than or equal to zero when x is in the interval (-∞, -3/2] U [4, ∞).