Problem 60 - College Algebra

Question

Solve \(x^{2}-x>12\)

Step-by-Step Solution

Answer

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

Step 1: Rewrite the Inequality

Start with the given inequality: \(x^2 - x > 12\). To make it easier to solve, subtract 12 from both sides of the inequality. This gives us \(x^2 - x - 12 > 0\).

Step 2: Factor the Quadratic Expression

The quadratic expression \(x^2 - x - 12\) needs to be factored. Look for two numbers that multiply to -12 and add to -1. These numbers are -4 and 3. So, the factored form is \((x - 4)(x + 3) > 0\).

Step 3: Identify Critical Points

The factors \((x - 4)\) and \((x + 3)\) are zero at \(x = 4\) and \(x = -3\), respectively. These are the critical points that divide the number line into intervals.

Step 4: Test the Intervals

The critical points \(x = 4\) and \(x = -3\) divide the number line into three intervals: \((-\infty, -3)\), \((-3, 4)\), and \((4, \infty)\). Choose a test point from each interval to determine where the inequality \((x - 4)(x + 3) > 0\) holds:- For \((-\infty, -3)\), choose \(x = -4\): \((-4 - 4)(-4 + 3) = ( -8)(-1) = 8 > 0\)- For \((-3, 4)\), choose \(x = 0\): \((0 - 4)(0 + 3) = (-4)(3) = -12 < 0\)- For \((4, \infty)\), choose \(x = 5\): \((5 - 4)(5 + 3) = (1)(8) = 8 > 0\)

Step 5: Write the Solution Set

The inequality \((x - 4)(x + 3) > 0\) holds true in intervals \((-\infty, -3)\) and \((4, \infty)\). Therefore, the solution set is \(x \in (-\infty, -3) \cup (4, \infty)\).

Key Concepts

Factoring Quadratics

To solve quadratic inequalities, understanding how to factor quadratic expressions is crucial. In our example, we started with the expression \(x^2 - x - 12\). To factor this quadratic, we needed to find two numbers that multiply to \(-12\) and add to \(-1\).
These numbers are \(-4\) and \(3\). This is because \(-4 \times 3 = -12\) and \(-4 + 3 = -1\).
Thus, we write the quadratic expression in its factored form: \((x - 4)(x + 3)\).

Factoring helps transform the quadratic inequality into a product of linear factors, making it easier to identify the solution. Factoring is all about breaking down the expression into simpler terms that can be used more effectively.

Inequality Solution

Solving an inequality means finding all the values of \(x\) that make the inequality true. After factoring, our inequality becomes \((x - 4)(x + 3) > 0\). The goal is to determine where this expression is greater than zero.

Here, the inequality is 'greater than' zero, which means we're looking for where the product \((x - 4)(x + 3)\) results in positive values. Understanding the inequality symbol and its significance is essential in identifying the graphical or numerical solution.

Critical Points

Critical points are the values of \(x\) that make each factor equal to zero. For the inequality \((x - 4)(x + 3) > 0\), these critical points are \(x = 4\) and \(x = -3\).

The critical points are important because they divide the number line into different intervals. Each of these intervals needs to be tested to see where the inequality holds true. Knowing the critical points helps in pinpointing areas on the graph where the function changes sign.

Interval Testing

Interval testing involves evaluating the inequality in different regions divided by the critical points. We identified two critical points: \(-3\) and \(4\), creating three intervals: \((-\infty, -3)\), \((-3, 4)\), and \((4, \infty)\).

For each interval, select a test point to see if the inequality \((x - 4)(x + 3) > 0\) is satisfied:

• In \((-\infty, -3)\), choosing \(x = -4\), results in a positive value, making the inequality true.


• In \((-3, 4)\), choosing \(x = 0\), results in a negative value, thus, the inequality is false.


• And, in \((4, \infty)\), choosing \(x = 5\), results in a positive value, indicating the inequality holds true.

Testing intervals ensures that you correctly identify the solution set where the original inequality is satisfied. This means the solution set for the inequality \((x - 4)(x + 3) > 0\) is \((-\infty, -3) \cup (4, \infty)\).