7x-6 greater than or equal to -3x^2 reduced and written in interval notation
7x - 6 ≤ - 3x^2
3x^2 + 7x - 6 ≤ 0
(x + 3)(3x - 2) ≤ 0
the zeros of the corresponding function are x = -3 and x = 2/3
so:
-3 ≤ x ≤ 2/3
convert to the notation that you use in your class, I prefer the notation I just used.
To solve the inequality 7x-6 ≥ -3x^2, we need to first rewrite it in the standard quadratic form. Here's how you can do it step by step:
Step 1: Move all the terms to one side of the inequality to obtain a quadratic expression equal to zero:
-3x^2 - 7x + 6 ≥ 0
Step 2: Factorize the quadratic expression if possible. In this case, the expression doesn't easily factorize, so we'll use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
The quadratic formula in this case is:
x = (7 ± √(7^2 - 4*(-3)*6)) / (2*(-3))
Simplifying further, we get:
x = (7 ± √(49 + 72)) / (-6)
x = (7 ± √121) / (-6)
x = (7 ± 11) / (-6)
Step 3: Solve for both possible values of x:
For x = (7 + 11) / (-6) = 18 / (-6) = -3,
and x = (7 - 11) / (-6) = -4 / (-6) = 2/3.
Step 4: Plot the solutions on a number line, which gives us the intervals:
-3 2/3
|---------------|---------------|
-3 2/3
Step 5: Determine whether the inequality is greater than or equal to zero based on the original inequality:
We have already considered the "equal to" part by including the endpoints in the number line above. Now, we need to determine whether the expression is positive or negative in the three intervals.
For the interval (-∞, -3), we choose a test point x = -4 and plug it into the expression:
-3x^2 - 7x + 6
-3(-4)^2 - 7(-4) + 6 = -48 + 28 + 6 = -14
Since the result is negative, the expression is less than zero in this interval.
For the interval (-3, 2/3), we choose a test point x = 0 and plug it into the expression:
-3(0)^2 - 7(0) + 6 = 6
Since the result is positive, the expression is greater than zero in this interval.
For the interval (2/3, ∞), we choose a test point x = 1 and plug it into the expression:
-3(1)^2 - 7(1) + 6 = -4
Since the result is negative, the expression is less than zero in this interval.
Step 6: Combine the intervals where the expression is greater than or equal to zero using interval notation:
The solution in interval notation is:
[-3, 2/3]