Question

A)(x+7)(x-1)<0 б)(x-2)(x+9)>

Answer 1

Объяснение:

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Answer 2

To solve the inequalities:

A) (x + 7)(x - 1) < 0 B) (x - 2)(x + 9) > 0

You can use the method of analyzing the signs of the factors and their combinations. To do this, you'll need to find the critical points where the factors become zero and then test intervals between these critical points to determine the sign of each expression.

Let's solve these inequalities step by step:

A) (x + 7)(x - 1) < 0

1. Find the critical points where the factors become zero:

   • Set each factor equal to zero and solve for x: x + 7 = 0 --> x = -7 x - 1 = 0 --> x = 1

2. Create intervals based on these critical points:

   • You have three intervals: (-∞, -7), (-7, 1), and (1, ∞).

3. Test a value in each interval to determine the sign of the expression:

   • Pick a value less than -7, such as x = -8. Plug it into the expression: (-8 + 7)(-8 - 1) = (-1)(-9) = 9, which is positive.
   • Pick a value between -7 and 1, such as x = 0. Plug it into the expression: (0 + 7)(0 - 1) = (7)(-1) = -7, which is negative.
   • Pick a value greater than 1, such as x = 2. Plug it into the expression: (2 + 7)(2 - 1) = (9)(1) = 9, which is positive.

4. Analyze the signs in the intervals:

   • In the interval (-∞, -7), the expression is positive.
   • In the interval (-7, 1), the expression is negative.
   • In the interval (1, ∞), the expression is positive.

Now, for inequality A), you want the expression to be less than zero:

• The expression is negative in the interval (-7, 1).
• So, the solution to A) is: -7 < x < 1.

B) (x - 2)(x + 9) > 0

1. Find the critical points where the factors become zero:

   • Set each factor equal to zero and solve for x: x - 2 = 0 --> x = 2 x + 9 = 0 --> x = -9

2. Create intervals based on these critical points:

   • You have three intervals: (-∞, -9), (-9, 2), and (2, ∞).

3. Test a value in each interval to determine the sign of the expression:

   • Pick a value less than -9, such as x = -10. Plug it into the expression: (-10 - 2)(-10 + 9) = (-12)(-1) = 12, which is positive.
   • Pick a value between -9 and 2, such as x = 0. Plug it into the expression: (0 - 2)(0 + 9) = (-2)(9) = -18, which is negative.
   • Pick a value greater than 2, such as x = 3. Plug it into the expression: (3 - 2)(3 + 9) = (1)(12) = 12, which is positive.

4. Analyze the signs in the intervals:

   • In the interval (-∞, -9), the expression is positive.
   • In the interval (-9, 2), the expression is negative.
   • In the interval (2, ∞), the expression is positive.

Now, for inequality B), you want the expression to be greater than zero:

• The expression is positive in the intervals (-∞, -9) and (2, ∞).
• So, the solution to B) is: x < -9 or x > 2.

To summarize: A) -7 < x < 1 B) x < -9 or x > 2