X(8-3x)+11<3(7-x)(x+7)+8

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вопрос Следующий
вопрос

To solve this inequality, we first need to simplify both sides of the equation.

Starting with the left side, we distribute the -3x to get:

8 - 3x + 11 < 3(7-x)(x+7) + 8
8 - 3x + 11 < 3(49 - 7x + 7x - x^2) + 8
8 - 3x + 11 < 3(49 - x^2) + 8
19 - 3x < 147 - 3x^2 + 8
19 - 3x < 155 - 3x^2

Now, we simplify the right side by expanding:

19 - 3x < 155 - 3x^2
19 - 3x < 155 - 3x^2

To further simplify, we rearrange the inequality to set it equal to 0 on the right side:

19 - 3x - 19 < 155 - 3x^2 - 19
-3x < -3x^2 + 136

Now we have a quadratic inequality in the form of -3x < -3x^2 + 136. To solve this, we first set it equal to zero:

-3x + 3x^2 - 136 > 0

Rearranging:

3x^2 - 3x - 136 < 0

We can further simplify by factoring the quadratic to find the critical points:

(3x + 17)(x - 8) < 0

The critical points are x = -17/3 and x = 8. Now we need to test each interval created by these critical points to find the solution set.

Testing the interval (-∞, -17/3), we pick a test point x = -4:

(3(-4) + 17)(-4 - 8) = (5)(-12) = -60

Since -60 is negative, this interval satisfies the inequality.

Testing the interval (-17/3, 8), we pick a test point x = 0:

(3(0) + 17)(0 - 8) = (17)(-8) = -136

Since -136 is negative, this interval satisfies the inequality.

Testing the interval (8, ∞), we pick a test point x = 10:

(3(10) + 17)(10 - 8) = (47)(2) = 94

Since 94 is positive, this interval does not satisfy the inequality.

Therefore, the solution set for the inequality is (-∞, -17/3) U (-17/3, 8).