Question

In Problems \(7-22,\) solve each inequality. 8\. \(x^{2}+3 x-10>0\)

Short answer

The solution is (-∞, -5) ∪ (2, ∞).

Step-by-step solution

- Factor the Quadratic Expression

To solve the inequality, start by factoring the quadratic expression on the left side of the inequality. We need to find two numbers that multiply to -10 and add up to 3. Notice that -2 and 5 satisfy these conditions. Thus, the expression factors as follows: (x+5)(x-2)>0.

- Determine the Critical Points

Set each factor equal to zero to determine the critical points. This gives us: x + 5 = 0 x = -5and x - 2 = 0 x = 2.

- Test Intervals Between Critical Points

The critical points divide the number line into three intervals: (-∞, -5), (-5, 2), and (2, ∞). Choose a test point in each interval and substitute it back into the factored inequality (x+5)(x-2)>0 to determine if the interval satisfies the inequality.

- Evaluate the Test Points

For the interval (-∞, -5), choose x = -6: (-6+5)(-6-2) = (-1)(-8) = 8 > 0. This interval satisfies the inequality. For the interval (-5, 2), choose x = 0: (0+5)(0-2) = (5)(-2) = -10 < 0. This interval does not satisfy the inequality. For the interval (2, ∞), choose x = 3: (3+5)(3-2) = (8)(1) = 8 > 0. This interval satisfies the inequality.

- Combine the Results

Since intervals (-∞, -5) and (2, ∞) satisfy the inequality, the solution set is: (-∞, -5) ∪ (2, ∞).

Key concepts

Factoring Quadratic Expressions

Quadratic expressions can often be factored into a product of two binomials. This is especially helpful in solving inequalities. For instance, the expression \( x^2 + 3x - 10 \) is factored by finding two numbers that multiply to -10 and add to 3. These numbers are -2 and 5, giving us the factors \( (x + 5)(x - 2) \). Factoring helps simplify the inequality, making it easier to find the solution.

Critical Points

The next step is to identify the critical points by setting each factor equal to zero. This gives us the equations \( x + 5 = 0 \) and \( x - 2 = 0 \). Solving these equations provides the critical points \( x = -5 \) and \( x = 2 \). These points divide the number line into different sections, which we will then test individually.

Test Intervals

With the critical points identified, the next task is to find the test intervals. These intervals are created by the critical points, \( -5 \text{ and } 2 \). This divides the number line into three segments: \( (-\infty, -5) \), \( (-5, 2) \), and \( (2, \infty) \). We then select test points from each interval to substitute back into the factored inequality to determine whether the interval satisfies the inequality.

Inequality Solution Set

Lastly, we evaluate the test points to compile the solution set. For \( (-\infty, -5) \), choosing \( x = -6 \) satisfies the inequality \( (x+5)(x-2) > 0 \). For \( (-5, 2) \), choosing \( x = 0 \) does not. For \( (2, \infty) \), choosing \( x = 3 \) again satisfies the inequality. Combining these results, the solution set for \( x^2 + 3x - 10 > 0 \) is \( (-\infty, -5) \cup (2, \infty) \).