Problem 8 - Algebra 2 and Trigonometry

Question

Write the solution set of each inequality if x is an element of the set of integers. \(x^{2}-8 x-20 \geq 0\)

Step-by-Step Solution

Answer

The solution set in integers is \( \{ ..., -4, -3, -2, 10, 11, 12, ... \} \).

Step 1: Identify the inequality

We are given the quadratic inequality \(x^2 - 8x - 20 \geq 0\) and must find integer solutions.

Step 2: Factor the quadratic expression

First, we solve the corresponding quadratic equation \(x^2 - 8x - 20 = 0\) by factoring. Looking for two numbers whose product is \(-20\) and whose sum is \(-8\), we find \((x - 10)(x + 2) = 0\).

Step 3: Solve for critical points

Set each factor equal to zero: \(x - 10 = 0\) gives \(x = 10\) and \(x + 2 = 0\) gives \(x = -2\). These are the critical points.

Step 4: Test intervals defined by critical points

The critical points divide the number line into intervals: \((-\infty, -2)\), \([-2, 10]\), and \((10, \infty)\). Test a point in each interval in the inequality \((x - 10)(x + 2) \geq 0\).

Step 5: Test interval \((-\infty, -2)\)

Choose \(x = -3\). Substituting, \((-3 - 10)(-3 + 2) = 13\cdot -1 = -13 < 0\). Thus, this interval does not satisfy the inequality.

Step 6: Test interval \([-2, 10]\)

Choose \(x = 0\). Substituting, \((0 - 10)(0 + 2) = -10 \cdot 2 = -20 < 0\). However, the endpoints \(x = -2\) and \(x = 10\) satisfy the equality, thus they are included.

Step 7: Test interval \((10, \infty)\)

Choose \(x = 11\). Substituting, \((11 - 10)(11 + 2) = 1\cdot 13 = 13 > 0\). So, this interval satisfies the inequality.

Step 8: State the solution set

The inequality \(x^2 - 8x - 20 \geq 0\) is satisfied for \(x \leq -2\) and \(x \geq 10\). Thus, the subset of integers is \(x \in \{ ..., -4, -3, -2, 10, 11, 12, ... \}\).

Key Concepts

Factoring Quadratics

To solve a quadratic inequality like \(x^2 - 8x - 20 \geq 0\), it's often helpful to factor the quadratic expression first. Factoring quadratics involves finding two numbers that multiply to give the constant term (here, \(-20\)) and add up to the linear coefficient (\(-8\)). Here are the steps in a simplified format:

• Identify the constant term and the linear coefficient.
• Find two numbers that multiply to the constant term and add to the linear coefficient.

By following these steps, we get \((x - 10)(x + 2) = 0\). This useful form allows us to identify critical points that are key to solving the inequality. Breaking down a quadratic expression into factors simplifies analyzing its behavior on a graph, especially around those critical points.

Critical Points

In the context of quadratic inequalities, critical points are what the factors of the quadratic equation equal. Once we have our quadratic equation factored as \((x - 10)(x + 2)\), setting each factor equal to zero gives us the critical points.

• For \(x - 10 = 0\), the solution is \(x = 10\).
• For \(x + 2 = 0\), the solution is \(x = -2\).

These values mark significant transitions on the number line. Understanding critical points is crucial because they show where the quadratic's graph crosses or touches the x-axis. In our inequality \(x^2 - 8x - 20 \geq 0\), the critical points are the boundaries for testing where the inequality holds true.

Number Line Testing Intervals

Once we know the critical points \(x = -2\) and \(x = 10\), they help us divide the number line into different intervals:

• \((-anity, -2)\)
• \([-2, 10]\)
• \((10, anity)\)

For each interval, we pick a test point and substitute it back into the inequality. This tells us whether the inequality holds in that region.Choosing a point in each interval makes the process of checking whether those intervals satisfy the inequality straightforward. If the inequality is satisfied (for example, it's \(\geq 0\) in that interval), then all values within that interval likely satisfy the inequality. This method ensures we account for the behavior of the quadratic function across its entire domain.

Integer Solutions

Finally, after determining which intervals satisfy the inequality \((x - 10)(x + 2) \geq 0\), we focus specifically on finding integer solutions. In our final solution,

• \((-anity, -2)\): does not satisfy the inequality.
• \([-2, 10]\): the endpoints are included, though the interior does not satisfy the inequality. Thus, integers here are \(-2\) and \(10\).
• (10, anity): satisfies the inequality. Meaning integers here are \(11, 12, 13,\) and so on.

We conclude that integer solutions for \(x^2 - 8x - 20 \geq 0\) include all integers from the set \{..., -4, -3, -2, 10, 11, 12, ...\}. Understanding which segments of the number line provide valid integer inputs helps students master solving quadratic inequalities, involving real-world applications like physics or economics.