Question

A)2x+4/x-7>0 b)x-1/x+5<=3 Решите неравенства

Answer 1

1.)от минус бесконечности до -2 , от 7 до плюс бесконечности.

1.)от минус бесконечности до -8 , от -5 до плюс бесконечности.

Answer 2

To solve the inequalities, we'll break them down into separate cases and find the solution intervals for each case. Remember that when dealing with rational inequalities, we need to consider the sign of both the numerator and the denominator separately.

a) 2x + 4 / (x - 7) > 0

Step 1: Find critical points by setting the numerator and denominator equal to zero and solving for x. Numerator: 2x + 4 = 0 2x = -4 x = -2

Denominator: x - 7 = 0 x = 7

Step 2: Create a sign chart to determine the intervals where the inequality is true.

Test Interval 1: x < -2 Pick x = -3 (a value less than -2) (2(-3) + 4) / (-3 - 7) = (-2) / (-10) = 1/5 > 0 (True)

Test Interval 2: -2 < x < 7 Pick x = 0 (a value between -2 and 7) (2(0) + 4) / (0 - 7) = 4 / (-7) < 0 (False)

Test Interval 3: x > 7 Pick x = 8 (a value greater than 7) (2(8) + 4) / (8 - 7) = (16 + 4) / 1 = 20 > 0 (True)

Step 3: Combine the intervals where the inequality is true.

The solution to the inequality 2x + 4 / (x - 7) > 0 is: x < -2 or x > 7.

b) (x - 1) / (x + 5) <= 3

Step 1: Find the critical point by setting the numerator equal to zero and solving for x. x - 1 = 0 x = 1

Step 2: Create a sign chart to determine the intervals where the inequality is true.

Test Interval 1: x < 1 Pick x = 0 (a value less than 1) (0 - 1) / (0 + 5) = -1 / 5 < 3 (True)

Test Interval 2: x = 1 (exclude from the solution due to the non-strict inequality)

Test Interval 3: x > 1 Pick x = 2 (a value greater than 1) (2 - 1) / (2 + 5) = 1 / 7 < 3 (True)

Step 3: Combine the intervals where the inequality is true.

The solution to the inequality (x - 1) / (x + 5) <= 3 is: x < 1 or x > 1.

Since the critical point x = 1 is excluded from the solution (due to the non-strict inequality), the final solution is x < 1 or x > 1.