Problem 22 - Algebra 2

Question

Solve each inequality using a graph, a table, or algebraically. $$ x^{2}+3 x-28<0 $$

Step-by-Step Solution

Answer

The solution is \((-7, 4)\).

Step 1: Identify the Equation

Start by identifying the equation from the inequality. Here, the corresponding equation is \(x^2 + 3x - 28 = 0\).

Step 2: Factor the Quadratic Expression

To find the roots of the equation, try to factor the quadratic expression. Look for two numbers that multiply to \(-28\) and add to \(3\). These numbers are \(7\) and \(-4\). Thus, the quadratic can be factored as \((x + 7)(x - 4)=0\).

Step 3: Solve for the Roots

Set each factor to zero to find the roots: \(x + 7 = 0\) and \(x - 4 = 0\). Solving these gives the roots \(x = -7\) and \(x = 4\).

Step 4: Create Test Intervals

Use the roots to divide the number line into intervals: \((-\infty, -7)\), \((-7, 4)\), and \((4, \infty)\). These intervals will be used to test where the inequality holds true.

Step 5: Test Each Interval

Choose a test point from each interval to determine if the inequality is satisfied. Test with \(x = -8\) for \((-\infty, -7)\): \((-8)^2 + 3(-8) - 28 = 12 > 0\), so not a solution. Test with \(x = 0\) for \((-7, 4)\): \(0^2 + 3(0) - 28 = -28 < 0\), so this is a solution. Test with \(x = 5\) for \((4, \infty)\): \(5^2 + 3(5) - 28 = 12 > 0\), so not a solution.

Step 6: Determine the Solution Set

The solution is the interval \((-7, 4)\) where the inequality \(x^2 + 3x - 28 < 0\) holds true. Therefore, \(x\) is in the open interval \((-7, 4)\).

Key Concepts

Factoring Quadratic Expressions

Factoring quadratic expressions is a powerful tool for solving quadratic inequalities. It involves expressing the quadratic equation in the form

• ax² + bx + c
• as a product of two binomials.

To factor a quadratic expression, you need to find two numbers that multiply to the constant term (here it is -28 ) and add to the linear coefficient (in this case 3 ).

For example, in the inequality x² + 3x - 28 < 0, the numbers 7 and -4 were identified because

• 7 × (-4) = -28
• and 7 + (-4) = 3.

Using these numbers, you can factor the expression into

• (x + 7)(x - 4)

.

Factoring not only simplifies the equation but also helps in directly finding the roots. It transforms the problem into a more manageable one by breaking it down into linear factors.

Roots of Quadratic Equations

The roots of a quadratic equation like

• x² + 3x - 28 = 0,

are the values of x that make the equation true when set to zero.

They represent the points where the graph of the quadratic equation intersects the x-axis. To find these roots, once the quadratic expression is factored, each factor is set to zero:

• x + 7 = 0
• and x - 4 = 0.

Solving these equations gives the roots

• x = -7
• and x = 4.

Roots are critical for sketching graphs and determining the solution intervals for inequalities. By identifying x = -7 and x = 4 as roots, you can divide the number line into testable intervals that reveal where the inequality holds true.

Test Intervals

Test intervals are regions on the number line divided by the roots of the quadratic equation.

After finding the roots of the equation, these values create boundaries for testing where the inequality is true. For the quadratic inequality x² + 3x - 28 < 0, using the roots x = -7 and x = 4, you create:

• Interval (-∞, -7)
• Interval (-7, 4)
• Interval (4, ∞)

By choosing any number within each interval (called a test point), you determine whether the inequality is satisfied in that interval.

- Testing x = -8 in (-∞, -7) shows the inequality is false. - Testing x = 0 in (-7, 4) shows the inequality is true. - Testing x = 5 in (4, ∞) shows it is false again.

The solution to the inequality is the interval (-7, 4) , where the quadratic expression is less than zero.