Solve give your answer in interval notation: 5-4x^2>=8x!!! Please help ASAP!!!:(
0 ≥ 4 x^2 + 8 x - 5
0 ≥ (2 x + 5) (2 x - 1)
for their product to be positive (>0), both factors must have the same sign
both ≥ 0 ... x ≥ 1/2
both ≤ 0 ... x ≤ -5/2
5-4x^2 ≥ 8x
-4x^2 - 8x + 5 ≥ 0
4x^2 + 8x - 5 ≤ 0
(2x + 5)(2x - 1) ≤ 0
Where is the parabola y = (2x+5)(2x-1) below the x-axis ?
-5/2 ≤ x ≤ 1/2
https://www.wolframalpha.com/input/?i=5-4x%5E2-+8x%E2%89%A50
To solve the inequality 5 - 4x^2 ≥ 8x, we will follow these steps:
Step 1: Rewrite the inequality in standard form, so that the coefficient of the highest power of x is positive:
4x^2 + 8x - 5 ≤ 0
Step 2: Next, we need to find the x-values where the expression on the left-hand side of the inequality is equal to zero. We can solve for x by factoring, completing the square, or using the quadratic formula. In this case, factoring is the simplest method:
(2x - 1)(2x + 5) ≤ 0
Step 3: Set each factor equal to zero and solve for x:
2x - 1 = 0 --> 2x = 1 --> x = 1/2 (or 0.5)
2x + 5 = 0 --> 2x = -5 --> x = -5/2 (or -2.5)
Step 4: Create a number line and plot the x-values we found in step 3:
... -3 -2.5 -1 -0.5 0 0.5 1 2 3 ...
Step 5: To determine the sign of the inequality in each interval, test any x-value within each interval. For simplicity, we'll use "-1" for the intervals to the left of -2.5, "0" for the interval between -2.5 and 0.5, and "1" for the intervals to the right of 0.5.
For x = -1:
(2(-1) - 1)(2(-1) + 5) = (-3)(3) = -9 (negative)
For x = 0:
(2(0) - 1)(2(0) + 5) = (-1)(5) = -5 (negative)
For x = 1:
(2(1) - 1)(2(1) + 5) = (1)(7) = 7 (positive)
Step 6: Now, we can write down the solution in interval notation based on the signs we found in step 5. We include the x-values where the expression equals zero as well:
(-∞, -2.5] ∪ (0.5, ∞)
So, the solution to the inequality 5 - 4x^2 ≥ 8x is (-∞, -2.5] ∪ (0.5, ∞) in interval notation.