Question

Construct a graphical representation of the inequality \(\mathrm{x}^{2}-2 \mathrm{x}-8 \leq 0\) and identify the solution set.

Short answer

The solution set for the inequality \(x^2 - 2x - 8 \leq 0\) is \(-2 < x < 4\) or, in interval notation, \((-2, 4)\). The graph is a segment on the x-axis between these values.

Step-by-step solution

Find the Roots of the Quadratic Equation

First, we need to find the roots of the equation \(x^2 - 2x - 8 = 0\). To do that, we can use the quadratic formula: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) In this case, \(a = 1\), \(b = -2\), and \(c = -8\) so, \(x = \frac{2 \pm \sqrt{(-2)^2 - 4(1)(-8)}}{2(1)}\) \(x = \frac{2 \pm \sqrt{36}}{2}\) Now, we get the two roots: \(x_1 = \frac{2 + 6}{2} = 4\) \(x_2 = \frac{2 - 6}{2} = -2\)

Test the Intervals

Now that we have the roots of the quadratic equation, we'll test the intervals formed by these roots to determine where the inequality is less than or equal to zero. 1. For \(x < -2\), choose a test point such as \(x = -3\): \((-3)^2 - 2(-3) - 8 = 9 + 6 - 8 = 7 > 0\), so the inequality is false. 2. For \(-2 < x < 4\), choose a test point such as \(x = 0\): \((0)^2 - 2(0) - 8 = -8 \leq 0\), so the inequality is true. 3. For \(x > 4\), choose a test point such as \(x = 5\): \((5)^2 - 2(5) - 8 = 25 - 10 - 8 = 7 > 0\), so the inequality is false.

Graph the Solution

Now, we'll graph the solution on the coordinate plane. Since the inequality is true for \(-2 < x < 4\), the solution set includes all values between -2 and 4, and the graph is a segment on the x-axis between these values.

Write the Solution Set

Finally, we'll write down the solution set for the inequality: The solution set for the inequality \(x^2 - 2x - 8 \leq 0\) is \(-2 < x < 4\) or, in interval notation, \((-2, 4)\).

Key concepts

Quadratic Inequalities

Quadratic inequalities such as \(x^2 - 2x - 8 \leq 0\) can often introduce a challenge because they're not as straightforward as linear inequalities. They involve a quadratic expression and a relational operator like \(\leq\), \(\geq\), \(>\), or \(<\). To solve them, we must first find the roots of the associated quadratic equation, which in this example is \(x^2 - 2x - 8 = 0\). These roots split the x-axis into intervals that we must test separately to determine where the inequality holds true.

Essentially, we're looking for the set of x-values that make the quadratic expression non-positive (\(\leq 0\)) or non-negative (\(\geq 0\)), depending on the inequality. When graphing the solutions, we can shade the regions of the x-axis that satisfy the inequality and often use a solid line to include the roots if the relational operator includes equality (\(\leq\) or \(\geq\)).

Quadratic Formula

The quadratic formula is used to find the roots of quadratic equations like \(ax^2 + bx + c = 0\). It states that the roots can be found using \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).

This powerful formula allows us to solve any quadratic equation, even when factoring is not possible. For our inequality example \(x^2 - 2x - 8 \leq 0\), we use the quadratic formula by setting \(a=1\), \(b=-2\), and \(c=-8\), to find the roots. Remember that \(\pm\) indicates two solutions; one with addition and one with subtraction. In our case, we obtain the roots as \(x_1 = 4\) and \(x_2 = -2\), which will be crucial in determining the solution set of the inequality.

Solution Set of Inequality

The solution set of an inequality refers to the range of values that satisfy the given inequality. In our quadratic inequality example, \(x^2 - 2x - 8 \leq 0\), we found the roots \(4\) and \( -2\), which divide the number line into distinct intervals.

By testing values within these intervals, we identify which ranges satisfy the inequality. For instance, the interval \(-2 < x < 4\) proves to satisfy our inequality, hence it becomes part of the solution set. In interval notation, this solution set is written as \( (-2, 4) \), illustrating the values between -2 and 4 that solve the inequality. This notation is compact and clearly articulates the range of solutions.

Graphing Inequalities on a Coordinate Plane

Graphing inequalities on a coordinate plane helps visualize the solution set. For our quadratic inequality \(x^2 - 2x - 8 \leq 0\), after determining the roots and the intervals where the inequality holds, we represent the solutions graphically.

We plot the roots \(4\) and \( -2\) on the x-axis and shade the region in between, since this is where the inequality is satisfied. If the inequality was strict (\( < \)) rather than inclusive (\( \leq \)), we'd use a dashed line instead of a solid one to denote the roots. Graphing not only provides a visual aid but also ensures a better understanding of the nature of the inequality's solutions.