Problem 12 - Algebra 2 and Trigonometry

Question

Write the solution set of each inequality if x is an element of the set of integers. \(x^{2}-x<6\)

Step-by-Step Solution

Answer

The solution set of the inequality is \(-1, 0, 1, 2\).

Step 1: Move all terms to one side

First, get a standard form for a quadratic inequality. Rearrange the inequality to have all terms on the same side: \( x^2 - x - 6 < 0 \).

Step 2: Solve the related quadratic equation

Find the roots of the quadratic equation \( x^2 - x - 6 = 0 \). Use the quadratic formula: \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a=1, b=-1, c=-6 \). The discriminant is \( (-1)^2 - 4(1)(-6) = 1 + 24 = 25 \). So, the roots are \( x = \frac{1 \pm 5}{2} \). This gives roots at \( x = 3 \) and \( x = -2 \).

Step 3: Determine test intervals

Use the roots found to divide the number line into intervals: \((-inity, -2), (-2, 3), (3, inity)\). We now test values from each interval to determine where the inequality holds.

Step 4: Test intervals

Choose a test point from each interval:- For \((-inity, -2)\), test \( x = -3 \): \( (-3)^2 - (-3) - 6 = 9 + 3 - 6 = 6 \), not less than 0.- For \((-2, 3)\), test \( x = 0 \): \( 0^2 - 0 - 6 = -6 \), which is less than 0.- For \((3, inity)\), test \( x = 4 \): \( 4^2 - 4 - 6 = 16 - 4 - 6 = 6 \), not less than 0.

Step 5: Write the solution set considering integers

The inequality \( x^2 - x < 6 \) holds for the interval \((-2, 3)\), where \(0\) is negative. Since \(x\) must be an integer, the solution set is the integers within the interval, \(-1, 0, 1, 2 \).

Key Concepts

Understanding the Quadratic Formula

When dealing with quadratic inequalities, one of the first steps is to solve the related quadratic equation. This is where the quadratic formula comes into play. The formula helps us find the roots (or solutions) of any quadratic equation of the form \( ax^2 + bx + c = 0 \).
The quadratic formula is:

• \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)

Here, \( \pm \) indicates that there are generally two roots: one found by adding the square root term and another by subtracting it.
Let's break down the formula:

• \( b^2 - 4ac \) is called the discriminant. It determines the nature of the roots.
• If the discriminant is positive, there are two real and distinct roots.
• If it's zero, there's exactly one real root.
• If it's negative, there are no real roots, only complex ones.

In our specific exercise, after calculating the discriminant \( (25) \) and applying the formula, we find the roots to be \( x = 3 \) and \( x = -2 \). These roots are crucial for determining the intervals we'll test next.

Defining the Solution Set

Defining the solution set of a quadratic inequality involves identifying all the values of \( x \) where the inequality holds true. In this context, the inequality we have is:\[ x^2 - x < 6 \]
This inequality does not necessarily hold true for all numbers, so we need to specify for which values of \( x \) it is valid.

• After simplifying, we have \( x^2 - x - 6 < 0 \).
• The solution involves solving for \( x \) in the real number line.

From our step-by-step solution, once we've identified the critical point roots \( x = 3 \) and \( x = -2 \), we then identify the intervals on the number line that will form our potential solution set.
The challenge is to find the right interval(s) where the inequality holds true, which will guide us toward defining the actual solution set.

Understanding Inequality Intervals

One of the critical parts of solving quadratic inequalities is determining the inequality intervals. These intervals help us check where the inequality, such as \( x^2 - x - 6 < 0 \), is true.

• After finding the roots of the related quadratic equation, divide the number line using these roots.
• The roots \( x = -2 \) and \( x = 3 \) create our intervals, which are: \((-\infty, -2), (-2, 3), (3, \infty)\).

The goal is to test each interval to determine where the inequality \( x^2 - x - 6 < 0 \) is valid.
Intervals that include the solution will be part of our solution set, so testing them methodically is crucial. This step ensures that we are thorough and accurate, therefore allowing us to pin down the solution set accurately for our particular problem.

Importance of Testing Intervals

To find the solution set for the inequality, we perform testing intervals. Testing intervals involve the selection of a test point from within each interval and substituting it back into the inequality.

• For interval \((-\infty, -2)\), testing with \( x = -3 \) resulted in a positive value, which does not satisfy our inequality.
• For interval \((-2, 3)\), using \( x = 0 \) resulted in a negative value, which satisfies the inequality.
• For interval \((3, \infty)\), testing with \( x = 4 \) also produced a positive value, which does not meet the inequality criteria.

Only the interval \((-2, 3)\) satisfied the condition \( x^2 - x - 6 < 0 \) after testing. However, since we are looking for integer solutions, we only include integer values within this interval.
Ultimately, when considering integer values, the solution set for \( x \) becomes \(-1, 0, 1, \text{ and } 2\). This process is essential for isolating the correct solutions in a structured way, ensuring accuracy and reliability in your results.