Question

2*x*2-1>=8решить неравенство

Answer 1

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Answer 2

To solve the inequality 2x^2 - 1 >= 8, follow these steps:

Step 1: Subtract 8 from both sides of the inequality: 2x^2 - 1 - 8 >= 0

Step 2: Simplify the left side of the inequality: 2x^2 - 9 >= 0

Step 3: Factor the quadratic expression, if possible. However, in this case, the expression cannot be factored.

Step 4: Find the critical points by setting the expression equal to zero and solving for x: 2x^2 - 9 = 0

Now, you can solve for x by using the quadratic formula: x = (-b ± √(b^2 - 4ac)) / 2a

In our case, a = 2, b = 0, and c = -9: x = (± √(0^2 - 4 * 2 * -9)) / 2 * 2 x = (± √(0 + 72)) / 4 x = (± √72) / 4

Step 5: Simplify the expression further: x = ± √(72) / 4 x = ± √(36 * 2) / 4 x = ± (6 * √2) / 4 x = ± (3 * √2) / 2

Step 6: Now, let's determine the intervals where the inequality is true.

a) For x < - (3 * √2) / 2: Plug in a value less than -(3 * √2) / 2 into the inequality, e.g., x = -2: 2(-2)^2 - 1 >= 8 2 * 4 - 1 >= 8 8 - 1 >= 8 7 >= 8

Since 7 is not greater than or equal to 8, this interval is NOT part of the solution.

b) For x = - (3 * √2) / 2: Plug in x = -(3 * √2) / 2 into the inequality: 2(- (3 * √2) / 2)^2 - 1 >= 8 2(9 * 2) / 4 - 1 >= 8 18/4 - 1 >= 8 9/2 - 1 >= 8 7/2 >= 8

Again, the inequality is not true for this value.

c) For -(3 * √2) / 2 < x < (3 * √2) / 2: We can see that in this interval, the expression 2x^2 - 9 will be positive, so it will satisfy the inequality.

d) For x = (3 * √2) / 2: Similar to before, this value does not satisfy the inequality.

e) For x > (3 * √2) / 2: Again, for values greater than (3 * √2) / 2, the expression 2x^2 - 9 will be positive, so it will satisfy the inequality.

Step 7: Final Solution The solution to the inequality 2x^2 - 1 >= 8 is the interval: -(3 * √2) / 2 < x < (3 * √2) / 2