Question

Розв’яжіть нерівність: 1) -3x < 12 2) 4x - x2 > 0

Answer 1

Відповідь:

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Пояснення:

Answer 2

1. -3x < 12

To solve this inequality, we'll divide both sides by -3, but we need to remember that when we multiply or divide an inequality by a negative number, we must reverse the inequality sign.

-3x < 12

Divide both sides by -3, remembering to flip the inequality sign:

x > 12 / -3

x > -4

The solution to the inequality is x > -4.

2. 4x - x^2 > 0

To solve this inequality, we'll first find the critical points by setting the expression equal to zero:

4x - x^2 = 0

Rearranging the terms:

x^2 - 4x = 0

Factorizing:

x(x - 4) = 0

This equation is true when either x = 0 or x - 4 = 0.

So the critical points are x = 0 and x = 4.

Now, we'll examine the sign of the expression in each interval created by these critical points.

For x < 0: Choose a value less than 0, such as x = -1: 4(-1) - (-1)^2 = -4 - 1 = -5, which is negative.

For 0 < x < 4: Choose a value between 0 and 4, such as x = 2: 4(2) - (2)^2 = 8 - 4 = 4, which is positive.

For x > 4: Choose a value greater than 4, such as x = 5: 4(5) - (5)^2 = 20 - 25 = -5, which is negative.

From the above analysis, we can see that the expression 4x - x^2 is positive when 0 < x < 4.

Therefore, the solution to the inequality is 0 < x < 4.