Problem 44 - Precalculus Student Solutions Manual 5th

Question

Solve each quadratic inequality. Write each solution set in interval notation. $$-x^{2}-6 x-16>-8$$

Step-by-Step Solution

Answer

(-∞, -4) ∪ (-4, -2)

Step 1: Move all terms to one side of the inequality

First, we need to move all terms to one side of the inequality so that the other side is zero. In this case, add 8 to both sides of the inequality:o(-x^2 - 6x - 16 + 8 > -8 + 8o(-x^2 - 6x - 8 > 0o- (x^2 + 6x + 8) > 0oNow the inequality is in the standard form for quadratic inequalities, which is: -(x^2 + 6x + 8) > 0.

Step 2: Solve the quadratic equation

We need to find the roots of the quadratic equation -(x^2 + 6x + 8) = 0. First, move the negative sign:(x^2 + 6x + 8 = 0)Factorizing the quadratic expression:(x + 2)(x + 4) = 0. So, the solutions are x = -2 and x = -4.

Step 3: Determine the intervals to test

The critical points divide the number line into three intervals: (-∞, -4), (-4, -2), and (-2, ∞).We will check the sign of the expression in each interval by choosing a test point from each interval.

Step 4: Test the intervals

Choose a test point in each interval and substitute into the expression -(x^2 + 6x + 8):For the interval (-∞, -4), choose x = -5 -(x^2 + 6x + 8) = -[(-5)^2 + 6(-5) + 8] = -(25 - 30 + 8) = -(-3) = 3 (positive).For the interval (-4, -2), choose x = -3 -(x^2 + 6x + 8) = -[(-3)^2 + 6(-3) + 8] = -(9 - 18 + 8) = -(-1) = 1 (positive).For the interval (-2, ∞), choose x = -1 -(x^2 + 6x + 8) = -[(-1)^2 + 6(-1) + 8] = -(1 - 6 + 8) = -(3) = -3 (negative).

Step 5: Determine the solution set

From the test points, we find that the inequality -(x^2 + 6x + 8) > 0 holds in the intervals (-∞, -4) and (-4, -2), which means our solution set is x < -2 and x < -4.

Step 6: Write the solution in interval notation

The solution set in interval notation is (-∞, -4) ∪ (-4, -2).

Key Concepts

solving quadratic inequalities

When solving quadratic inequalities, the goal is to determine the set of values that satisfy the inequality. It helps to follow a systematic approach. Start by rearranging the inequality so that one side is zero. Then solve the corresponding quadratic equation to find critical points, which divide the number line into intervals. By testing points within these intervals, you determine where the inequality holds. This method provides a structured way to find the solution set.

interval notation

Interval notation is a method used to represent the set of solutions for inequalities. It uses brackets and parentheses to indicate the range of values. For example:
- \((- ∞, -4) \) means all values less than -4.
- \((-4, -2) \) represents all values between -4 and -2, but not including -4 and -2.
- ∪ symbolizes the union of multiple intervals. For example, \((- ∞, -4) ∪ (-4, -2) \) combines both intervals, representing values less than -2. To effectively use interval notation, knowing inequality symbols and how to manage boundary values is essential.

factoring quadratics

Factoring quadratics is a key step in solving equations and inequalities. The method involves expressing a quadratic equation \((ax^2 + bx + c) \) as a product of two binomials, \((m + n)(p + q) \). For example, for \((x^2 + 6x + 8) \), factor to get \((x + 2)(x + 4) \). This reveals the roots: \((-2) \) and \((-4) \). These roots are critical in determining intervals on the number line. Factoring simplifies the process, making it easier to solve the equation.

number line method

The number line method visualizes intervals and tests points within them to confirm where the inequality holds true. After finding critical points (roots), divide the number line into separate intervals. Choose a test point from each interval, substitute it back into the original inequality, and observe the sign. If the resulting value satisfies the inequality, the interval belongs to the solution set. This graphical representation is practical and simplifies understanding. The segments on the number line directly show where the quadratic inequality's solution lies.