Problem 22 - Algebra for College Students

Question

Solve each inequality. $$x^{2}+3 x-54<0$$

Step-by-Step Solution

Answer

The solution is \((-9, 6)\).

Step 1: Understand the Inequality

The inequality is a quadratic function, which means it represents a parabola when graphed. Our goal is to find the values of \(x\) for which the parabola is below the \(x\)-axis.

Step 2: Factor the Quadratic

We need to factor the quadratic expression \(x^2 + 3x - 54\). This can be factored into two binomials: \[x^2 + 3x - 54 = (x + 9)(x - 6)\] Verify by expanding the factors to ensure they multiply back to the original quadratic expression.

Step 3: Find the Roots of the Quadratic Equation

Set the equation \((x + 9)(x - 6) = 0\) to find the roots of the quadratic equation. Solving this gives us the roots:\[x + 9 = 0 \quad \Rightarrow \quad x = -9\]\[x - 6 = 0 \quad \Rightarrow \quad x = 6\]These roots divide the number line into intervals.

Step 4: Test Intervals on the Number Line

The roots \(-9\) and \(6\) divide the number line into three intervals: 1. \((-\infty, -9)\)2. \((-9, 6)\)3. \((6, \infty)\)Select test points from each interval to determine where the inequality holds true (i.e., where the product of the factors is negative).

Step 5: Evaluate the Test Intervals

Choose test points: - For \((-\infty, -9)\): pick \(x = -10\)- For \((-9, 6)\): pick \(x = 0\)- For \((6, \infty)\): pick \(x = 7\)Evaluate at each test point:- \((x + 9)(x - 6)\) at \(x = -10\): \((-10 + 9)(-10 - 6) = (-1)(-16) = 16 > 0\)- \((x + 9)(x - 6)\) at \(x = 0\): \((0 + 9)(0 - 6) = (9)(-6) = -54 < 0\)- \((x + 9)(x - 6)\) at \(x = 7\): \((7 + 9)(7 - 6) = (16)(1) = 16 > 0\)

Step 6: Determine the Solution

Based on the test intervals, we conclude that the inequality \((x + 9)(x - 6) < 0\) holds true for the interval \((-9, 6)\). Thus, the solution to the inequality is the values of \(x\) between -9 and 6 (exclusive), which is:\((-9, 6)\).

Key Concepts

Quadratic Inequalities

In mathematics, a quadratic inequality is similar to a quadratic equation but instead of an equal sign, it uses inequality symbols like <, >, ≤, or ≥. These inequalities represent a range of values for which the inequality holds true. When solving quadratic inequalities such as \(x^2 + 3x - 54 < 0\), you're fundamentally looking to find where the graph of the quadratic equation lies below the x-axis. This is because the inequality '<' signifies that the resulting value should be less than zero. Quadratic inequalities are more nuanced than linear inequalities because they can represent a curve, and their solutions come in ranges rather than singular values.

Factoring Quadratics

Factoring is a key technique used to simplify and solve quadratic expressions. For a given quadratic inequality, like \(x^2 + 3x - 54 < 0\), the first step is often to factor the quadratic equation. This means writing it in the form of two binomials.

In this case, \(x^2 + 3x - 54\) is factored as \((x + 9)(x - 6)\). To do this, one must find two numbers that multiply to give the constant term (-54), and add up to give the coefficient of the linear term (3). After finding these numbers, the quadratic can be rewritten as a product of two binomials.

• Verify your factorization by expanding it back to ensure it matches the original quadratic expression.

This factorization helps us identify the 'roots' of the associated quadratic equation.

Number Line Analysis

Number line analysis is a crucial part of solving quadratic inequalities. Once you have the factored form of a quadratic, the next step is to determine where the inequality holds by examining the intervals around its roots.

Consider the roots \(x = -9\) and \(x = 6\). They divide the number line into three intervals: \((-fty, -9)\), \((-9, 6)\), and \((6, \infty)\). The solution to the quadratic inequality involves determining which intervals satisfy the condition \((x + 9)(x - 6) < 0\). At each interval, a test point is chosen to determine the sign of the quadratic expression within that segment. This helps to clarify which sections of the number line meet the inequality criteria.

Test Interval Method

The test interval method is an effective way of solving quadratic inequalities, especially once the quadratic is factored and its roots are identified.

The basic idea behind this method is to select a point in each interval created by the roots of the quadratic to check the sign of the expression. For example, when you have the factored inequality \((x + 9)(x - 6) < 0\):

• Choose a point like \(x = -10\) in the interval \((-fty, -9)\), \(x = 0\) in \((-9, 6)\), and \(x = 7\) in \((6, \infty)\).
• Substitute each chosen point into the inequality, and check the sign of the resulting expression.

This helps determine where the expression is negative—a requirement for the solution to the quadratic inequality. Remember, the test interval method simplifies solving inequalities by reducing the problem to checking a few specific values.