(2x+1)(x-3)(x+7) 0

Jasonbradley

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Jun 16, 2024

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Solving the Inequality: (2x+1)(x-3)(x+7) < 0

This inequality involves a product of three linear factors. To solve it, we need to find the intervals where the expression is negative. Here's how we can approach it:

1. Find the Critical Points

The critical points are the values of x that make the expression equal to zero. So, set each factor equal to zero and solve:

• 2x + 1 = 0 => x = -1/2
• x - 3 = 0 => x = 3
• x + 7 = 0 => x = -7

These critical points divide the number line into four intervals:

• Interval 1: x < -7
• Interval 2: -7 < x < -1/2
• Interval 3: -1/2 < x < 3
• Interval 4: x > 3

2. Test Each Interval

We need to determine the sign of the expression (2x+1)(x-3)(x+7) in each interval:

Interval 1 (x < -7):

• 2x+1: Negative
• x-3: Negative
• x+7: Negative
• Product: Negative * Negative * Negative = Negative

Interval 2 (-7 < x < -1/2):

• 2x+1: Negative
• x-3: Negative
• x+7: Positive
• Product: Negative * Negative * Positive = Positive

Interval 3 (-1/2 < x < 3):

• 2x+1: Positive
• x-3: Negative
• x+7: Positive
• Product: Positive * Negative * Positive = Negative

Interval 4 (x > 3):

• 2x+1: Positive
• x-3: Positive
• x+7: Positive
• Product: Positive * Positive * Positive = Positive

3. Solution

The inequality (2x+1)(x-3)(x+7) < 0 is satisfied when the expression is negative. Therefore, the solution is:

x < -7 or -1/2 < x < 3

We can represent this solution on a number line:

   <-----|-----|-----|----->
       -7   -1/2  3

The open circles indicate that the critical points are not included in the solution, since the inequality is strictly less than zero.