Problem 7 - Algebra 2

Question

Solve each inequality using a graph, a table, or algebraically. $$ x^{2}-x-12>0 $$

Step-by-Step Solution

Answer

The solution is \((-\infty, -3) \cup (4, \infty)\).

Step 1: Find the Roots of the Equation

To solve the inequality, first find the values of \(x\) where the quadratic equation \(x^2 - x - 12 = 0\) equals zero. Use the quadratic formula or factor the expression. Factoring gives \((x - 4)(x + 3) = 0\). So, the roots are \(x = 4\) and \(x = -3\).

Step 2: Determine the Intervals

The roots \(x = 4\) and \(x = -3\) divide the real number line into three intervals: \((-\infty, -3)\), \((-3, 4)\), and \((4, \infty)\). We will test the sign of the inequality in each of these intervals.

Step 3: Test the Intervals for the Inequality

Choose a test point from each interval and substitute it into the inequality \(x^2 - x - 12 > 0\). For \((-\infty, -3)\), choose \(x = -4\): \((-4)^2 - (-4) - 12 = 4 > 0\). For \((-3, 4)\), choose \(x = 0\): \(0^2 - 0 - 12 = -12 < 0\). For \((4, \infty)\), choose \(x = 5\): \(5^2 - 5 - 12 = 8 > 0\).

Step 4: Write the Solution

The inequality \(x^2 - x - 12 > 0\) is satisfied in the intervals \((-\infty, -3)\) and \((4, \infty)\). Therefore, these intervals are the solution to the inequality.

Key Concepts

Roots of Quadratic Equations

Quadratic equations often have two roots, which are values of \(x\) where the equation is satisfied. For our inequality \(x^2 - x - 12 > 0\), we first set up the equation \(x^2 - x - 12 = 0\) to find these roots.
Upon factoring, we find the equation can be expressed as \((x - 4)(x + 3) = 0\). This gives us two solutions for \(x\): \(x = 4\) and \(x = -3\). These solutions are known as the roots of the equation.
The roots help us partition the number line into sections that we will examine to see where the inequality holds true.

Factoring Quadratics

Factoring quadratics is one way to find the roots of quadratic equations. The process involves expressing a quadratic expression like \(ax^2 + bx + c\) as a product of two binomials. For \(x^2 - x - 12 = 0\), we look for two numbers that multiply to -12 and add to -1.
These numbers are -4 and 3, allowing us to factor the expression into \((x - 4)(x + 3)\). Thus, the quadratic expression is simplified into factors involving \(x\), which makes it easier to solve.
Factoring is useful because it provides a straightforward way to find where a quadratic equation or inequality equals zero, giving us key points (the roots) to analyze further.

Inequality Testing

With the roots from factoring, it’s time to test the inequality \(x^2 - x - 12 > 0\) in various intervals on the number line. The roots \(x = 4\) and \(x = -3\) divide the line into three distinct intervals:

• \((-\infty, -3)\)
• \((-3, 4)\)
• \((4, \infty)\)

For each of these intervals, we pick a test point that’s easy to work with.
- Choosing \(x = -4\) for \((-\infty, -3)\), we find \((-4)^2 - (-4) - 12 = 4 > 0\), meaning the inequality is satisfied.
- Testing \(x = 0\) for \((-3, 4)\), the result is \(0^2 - 0 - 12 = -12 < 0\), showing the inequality is not satisfied.
- Lastly, for \((4, \infty)\) with \(x = 5\), we find \(5^2 - 5 - 12 = 8 > 0\). Thus, this interval satisfies the inequality.
Therefore, the intervals where our inequality holds are \((-\infty, -3)\) and \((4, \infty)\).

Interval Notation

To express solutions to inequalities like \(x^2 - x - 12 > 0\) efficiently, we use interval notation. This system uses parentheses and brackets to describe a range of values.
In our example, the solution is where the inequality is true: these are the intervals \((-\infty, -3)\) and \((4, \infty)\). Parentheses \(()\) indicate that endpoints are not included in the solution, which applies here since \(x = -3\) and \(x = 4\) are where the inequality equals zero.
Interval notation provides a concise and clear way to represent the range of solutions, summarizing where an inequality holds true without needing lengthy descriptions.