Question

(2x-1)(2x+1)>2(X+0.5)2

Answer 1

Ответ: 3/2>0

Объяснение:

(2x+1)(2x-1)>2*(x+0.5)^2

4x^2-1>2*(x^2+x+0.25)

4x^2-1>2x^2+2x+0.5

2x^2-2x-1.5>0

В=(-2)^2-4*2*(-1.5)=4+12=16

x1=3/2

x2=-1/2

Answer 2

To solve the inequality (2x-1)(2x+1) > 2(x+0.5)^2, you can follow these steps:

1. Expand both sides of the inequality: (2x-1)(2x+1) > 2(x+0.5)^2

   Simplify: (4x^2 - 1) > 2(x^2 + x + 0.25)

2. Distribute the 2 on the right side: 4x^2 - 1 > 2x^2 + 2x + 0.5

3. Move all terms to the left side of the inequality to set it to zero: 4x^2 - 2x^2 - 2x - 1 - 0.5 > 0

4. Combine like terms: 2x^2 - 2x - 1.5 > 0

5. Now, we want to find the values of x that make this quadratic inequality true. You can use a sign chart or factor the quadratic to find the critical points. Factoring is a bit tricky in this case, so let's use a sign chart:

   a. Find the critical points by setting the expression inside the inequality to zero: 2x^2 - 2x - 1.5 = 0

   b. Solve for x: 2x^2 - 2x - 1.5 = 0 x^2 - x - 0.75 = 0

   You can factor the quadratic: (x - 1.5)(x + 0.5) = 0

   Set each factor equal to zero and solve for x: x - 1.5 = 0 => x = 1.5 x + 0.5 = 0 => x = -0.5

   So, the critical points are x = 1.5 and x = -0.5.

   c. Create a sign chart for the inequality: We have three intervals to consider: (-∞, -0.5), (-0.5, 1.5), and (1.5, ∞).

   Test a value in each interval: For x < -0.5, you can use x = -1. For -0.5 < x < 1.5, you can use x = 0. For x > 1.5, you can use x = 2.

   Plug these values into the inequality: -1: 2(-1)^2 - 2(-1) - 1.5 > 0 0: 2(0)^2 - 2(0) - 1.5 > 0 2: 2(2)^2 - 2(2) - 1.5 > 0

   d. Analyze the sign chart: In the interval (-∞, -0.5), the expression is positive. In the interval (-0.5, 1.5), the expression is negative. In the interval (1.5, ∞), the expression is positive.

6. Determine the solution: The inequality (2x-1)(2x+1) > 2(x+0.5)^2 is satisfied in the intervals where the expression is positive. So, the solution is:

   x < -0.5 or x > 1.5

Therefore, the solution to the inequality is the set of real numbers x that are less than -0.5 or greater than 1.5.