Question

Решение неравенства 2x^2≥x

Answer 1

(-бесконечности; 0] и [0,5; +бесконечности)

Answer 2

To solve the inequality 2x^2 ≥ x, we need to find the values of x that satisfy the inequality. Here's how you can do it step-by-step:

Step 1: Move all the terms to one side of the inequality to set it to zero: 2x^2 - x ≥ 0

Step 2: Factor out common terms (in this case, x): x(2x - 1) ≥ 0

Step 3: Find the critical points by setting each factor equal to zero and solving for x: x = 0 (from the factor x = 0) 2x - 1 = 0 ⇒ 2x = 1 ⇒ x = 1/2

Step 4: Now we have three intervals on the number line: (-∞, 0), (0, 1/2), and (1/2, ∞). Test a value within each interval to determine the sign of the expression within the interval.

For x < 0, let's use x = -1: 2(-1)^2 - (-1) = 2 - (-1) = 3 (positive)

For 0 < x < 1/2, let's use x = 1/4: 2(1/4)^2 - (1/4) = 2(1/16) - (1/4) = 1/8 - 1/4 = -1/8 (negative)

For x > 1/2, let's use x = 1: 2(1)^2 - (1) = 2 - 1 = 1 (positive)

Step 5: Analyze the signs in each interval:

• (-∞, 0): The expression 2x^2 - x is positive.
• (0, 1/2): The expression 2x^2 - x is negative.
• (1/2, ∞): The expression 2x^2 - x is positive.

Step 6: Determine the solution to the inequality:

Since the inequality is 2x^2 - x ≥ 0, we are looking for the values of x that make the expression non-negative (positive or zero).

Thus, the solution is the combination of intervals where the expression is non-negative: (-∞, 0] and [1/2, ∞).

In interval notation, the solution is: x ∈ (-∞, 0] ∪ [1/2, ∞).