Problem 22 - Beginning and Intermediate Algebra

Question

Solve each quadratic inequality. Graph the solution set and write the solution in interval notation. $$p^{2}-144>0$$

Step-by-Step Solution

Answer

The solution set to the quadratic inequality \(p^2 - 144 > 0\) is \((-∞, -12) \cup (12, \infty)\). The graph of the solution set can be represented as: ``` <---------O==========|==========O---------> -12 12 (p, +∞) (-∞, p) ```

Step 1: Rewrite the inequality

First, let's rewrite the inequality in the standard quadratic form so it's easier to work with. We have: \(p^2 - 144 > 0\)

Step 2: Find the critical points

To find the critical points, we need to find the values of \(p\) that make the inequality equal to zero. Set the inequality to an equality and solve: \(p^2 - 144 = 0\) To solve this quadratic equation, we can factor the difference of squares: \((p + 12)(p - 12) = 0\) This yields the critical points \(p = 12\) and \(p = -12\).

Step 3: Test intervals

Now, we'll test intervals to see which intervals satisfy the inequality. The critical points divide the number line into three intervals: \((-\infty, -12)\), \((-12, 12)\), and \((12, \infty)\). Test each interval by picking a representative value from each interval and plug into the original inequality. Interval 1: \((-\infty, -12)\) Choose \(p = -13\), then test in the inequality: \((-13)^2 - 144 > 0\) Of course, 169-144 > 0, so this interval satisfies the inequality. Interval 2: \((-12, 12)\) Choose \(p = 0\), then test in the inequality: \(0^2 - 144 > 0\) This yields -144 > 0, which is clearly false, so this interval does not satisfy the inequality. Interval 3: \((12, \infty)\) Choose \(p = 13\), then test in the inequality: \((13)^2 - 144 > 0\) Clearly, 169-144 > 0, so this interval satisfies the inequality.

Step 4: Graph the solution set

Now, we will graph the solution set by representing the intervals that satisfy the inequality on the number line. Since the critical points are not part of the solution (because the inequality is strict), we represent them as open circles on the number line. The intervals are shaded because they satisfy the inequality. Here's the graph: ``` <---------O==========|==========O---------> -12 12 (p, +∞) (-∞, p) ```

Step 5: Write the solution in interval notation

Finally, we will write the solution in interval notation using the intervals that satisfy the inequality: \((-∞, -12) \cup (12, \infty)\) So, the solution set to the quadratic inequality \(p^2 - 144 > 0\) is \((-∞, -12) \cup (12, \infty)\).

Key Concepts

Interval Notation

Interval notation helps us express the range of values that satisfy an inequality. It's like a shorthand for writing ranges.
For the quadratic inequality \(p^2 - 144 > 0\), the solution is written as \((-fty, -12) \cup (12, \infty)\).
This means all numbers less than -12 and all numbers greater than 12 are included.

• The round brackets \(()\) indicate that -12 and 12 are not included since the inequality is strict \( > 0\).
• The union symbol \( \cup \) connects the two separate intervals.

Understanding interval notation is important because it clearly communicates which parts of the number line are included in the solution.

Critical Points

Critical points help us determine where a quadratic inequality changes from true to false or vice versa.
For \(p^2 - 144 > 0\), setting \(p^2 - 144 = 0\) gives us the critical points.

• By factoring \((p + 12)(p - 12) = 0\), we find the critical points are \(p = 12\) and \(p = -12\).

These points divide the number line into intervals that you can test to see where the inequality holds true.
Understanding critical points can aid in visually identifying solution regions.

Quadratic Equation

Quadratic equations form the foundation for solving quadratic inequalities. They typically look like \(ax^2 + bx + c = 0\).
In our problem, the equation is \(p^2 - 144 = 0\).

Factoring

Factoring helps simplify the equation. Here, it's a difference of squares:

• \((p + 12)(p - 12) = 0\)

Solutions to a quadratic equation provide the critical points needed for solving inequalities.
Knowing how to solve quadratic equations is crucial for deeper understanding of inequalities.

Inequality Solution Graph

Graphing the solution set of a quadratic inequality helps visualize which values satisfy the inequality.
For \(p^2 - 144 > 0\), you'll plot specifically chosen test points.

• Open circles at \(-12\) and \(12\) indicate these points are not part of the solution because the inequality is strict.
• Shaded regions \((-fty, -12)\) and \((12, \infty)\) show where the inequality is true.

A well-drawn solution graph provides an immediate way to understand the solution behavior in relation to the number line.