Problem 114 - A Graphical Approach to Precalculus with Limits

Question

Solve each inequality analytically. Support your answers graphically. Give exact values for endpoints. (a) \(x^{2}+5 x<2\) (b) \(x^{2}+5 x \geq 2\)

Step-by-Step Solution

Answer

(a) Solution: \((-\infty, \frac{-5 - \sqrt{33}}{2}) \cup (\frac{-5 + \sqrt{33}}{2}, \infty)\); (b) Solution: \([\frac{-5 - \sqrt{33}}{2}, \frac{-5 + \sqrt{33}}{2}]\).

Step 1: Set the Inequality to Zero

To solve the inequality analytically, first convert it into an equation by setting it to 0. For part (a), start with the equation: \(x^2 + 5x - 2 = 0\).

Step 2: Solve the Quadratic Equation

Use the quadratic formula to find the roots of the equation. The quadratic formula is given by \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). Here, \(a = 1\), \(b = 5\), and \(c = -2\). Calculate the roots:\[x = \frac{-5 \pm \sqrt{(5)^2 - 4(1)(-2)}}{2(1)}\]\[x = \frac{-5 \pm \sqrt{25 + 8}}{2}\]\[x = \frac{-5 \pm \sqrt{33}}{2}\]

Step 3: Determine Intervals on the Number Line

Locate the roots \(x = \frac{-5 + \sqrt{33}}{2}\) and \(x = \frac{-5 - \sqrt{33}}{2}\) on the number line. These points divide the number line into three intervals: \((-\infty, \frac{-5 - \sqrt{33}}{2})\), \((\frac{-5 - \sqrt{33}}{2}, \frac{-5 + \sqrt{33}}{2})\), and \((\frac{-5 + \sqrt{33}}{2}, +\infty)\).

Step 4: Test Points in Each Interval

Choose a test point in each interval and substitute it into the inequality \(x^2 + 5x < 2\) for part (a).- Test a point in \((-\infty, \frac{-5 - \sqrt{33}}{2})\): Use \(x = -4\).- Test a point in \((\frac{-5 - \sqrt{33}}{2}, \frac{-5 + \sqrt{33}}{2})\): Use \(x = 0\).- Test a point in \((\frac{-5 + \sqrt{33}}{2}, +\infty)\): Use \(x = 3\).Check which intervals satisfy the inequality.

Step 5: Solve for the Other Inequality

Repeat steps 1 to 4 for part (b), solving \(x^2 + 5x \geq 2\). Use the same test intervals to determine where this inequality holds true.

Step 6: Graph the Solution

Graph the results for both inequalities to support your answers. For part (a), shade the region that satisfies \(x^2 + 5x < 2\) on a number line. For part (b), shade the region where \(x^2 + 5x \geq 2\).

Key Concepts

Quadratic Formula

The quadratic formula is a powerful tool used to solve quadratic equations, which are equations of the form \( ax^2 + bx + c = 0 \). This formula provides the solutions, also known as roots or zeros, of the quadratic equation. The formula is expressed as:

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

To use the quadratic formula correctly:

• Identify the coefficients: \( a\), \( b\), and \( c \).
• Plug these values into the formula.
• Solve for \( x \) to find the roots.

For an exercise with \( x^2 + 5x - 2 = 0 \):

• \( a = 1 \), \( b = 5 \), \( c = -2 \)
• Calculate \( b^2 - 4ac \).
• Find the square root and use it in the formula to solve for \( x \).

The roots you find are critical in determining the behavior of the quadratic inequality.

Graphical Representation

Graphing inequalities is an excellent way to visualize the solutions of an inequality. By plotting the solutions of the quadratic equation, we can better understand where the inequalities hold. First, plot the standard quadratic equation \( y = x^2 + 5x - 2 \) on a coordinate grid.Next, identify the roots on the x-axis. These roots were found using the quadratic formula. They are the points where the graph intersects the x-axis. This graph resembles a parabola which either opens upwards or downwards.

• It opens upwards when \( a > 0 \).
• It opens downwards when \( a < 0 \).

For the inequality \( x^2 + 5x < 2 \):

• Shade the area below the parabola to represent the solution set for this inequality.

For the inequality \( x^2 + 5x \geq 2 \):

• Shade the area on or above the parabola.These graphical solutions support the analytical work you have done in finding points that satisfy the inequalities.

Number Line Analysis

Number line analysis is a simple yet powerful method for solving inequalities. After calculating the roots of the equation, position these roots on a number line. This divides the number line into intervals, and we're interested in determining where the inequality holds.For example:

• The roots \( \frac{-5 + \sqrt{33}}{2} \) and \( \frac{-5 - \sqrt{33}}{2} \) provide division points.
• Mark these points on the number line, creating intervals: \(( -\infty, \frac{-5 - \sqrt{33}}{2} )\), \(( \frac{-5 - \sqrt{33}}{2}, \frac{-5 + \sqrt{33}}{2} )\), \(( \frac{-5 + \sqrt{33}}{2}, +\infty )\).

The process involves picking test points in these intervals to see where the original inequality is true. This helps isolate the regions where the inequality holds, giving a clear, visual depiction alongside the analytical approach.

Testing Intervals

Testing each interval is crucial to understanding which segments of the number line satisfy the inequality. This is done after using the quadratic formula and number line analysis.Select any point within each interval. Substitute this point back into the inequality.

• For the interval \(( -\infty, \frac{-5 - \sqrt{33}}{2} )\), test \( x = -4 \).
• For the interval \(( \frac{-5 - \sqrt{33}}{2}, \frac{-5 + \sqrt{33}}{2} )\), test \( x = 0 \).
• For the interval \(( \frac{-5 + \sqrt{33}}{2}, +\infty )\), test \( x = 3 \).

After testing:

• Determine whether the inequality is true or false for each test point.
• This confirms which intervals are part of your solution set.

Testing intervals ensures your solution is comprehensive and that all potential areas of the number line are considered.