Problem 108 - A Graphical Approach to Precalculus with Limits

Question

Solve each inequality analytically. Support your answers graphically. Give exact values for endpoints. (a) \(3 x^{2}+13 x \leq-10\) (b) \(3 x^{2}+13 x>-10\)

Step-by-Step Solution

Answer

(a) Solution: \([-\frac{10}{3}, -1]\) (b) Solution: \((-\infty, -\frac{10}{3}) \cup (-1, +\infty)\)

Step 1: Move All Terms to One Side (Inequality a)

Start by moving all the terms to one side of the inequality for part (a):\[3x^2 + 13x + 10 \leq 0\]This allows us to compare it directly to zero and identify it as a quadratic inequality.

Step 2: Find the Roots of the Quadratic Equation (Inequality a)

To find the critical points where the inequality might change, solve the quadratic equation:\[3x^2 + 13x + 10 = 0\]Use the quadratic formula, where \(a = 3\), \(b = 13\), and \(c = 10\):\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]\[x = \frac{-13 \pm \sqrt{169 - 120}}{6}\]\[x = \frac{-13 \pm \sqrt{49}}{6}\]\[x = \frac{-13 \pm 7}{6}\]The solutions are \(x = -1\) and \(x = -\frac{10}{3}\).

Step 3: Determine the Intervals (Inequality a)

With the critical points found, the real number line is divided into intervals:1. \((-\infty, -\frac{10}{3})\)2. \([-\frac{10}{3}, -1]\)3. \((-1, +\infty)\)Test a point from each interval to determine where the inequality holds true.

Step 4: Test Intervals (Inequality a)

Choose a test point from each interval:- For \((-\infty, -\frac{10}{3})\), choose \(-4\): \(3(-4)^2 + 13(-4) + 10 = 48 - 52 + 10 = 6 > 0\)- For \([-\frac{10}{3}, -1]\), choose \(-2\): \(3(-2)^2 + 13(-2) + 10 = 12 - 26 + 10 = -4 \leq 0\)- For \((-1, +\infty)\), choose \(0\): \(3(0)^2 + 13(0) + 10 = 10 > 0\)Thus, the inequality holds in the interval \([-\frac{10}{3}, -1]\).

Step 1: Move All Terms to One Side (Inequality b)

Now, solve part (b) by moving all the terms to one side:\[3x^2 + 13x + 10 > 0\]This setup looks at where the quadratic equation is greater than zero.

Step 2: Use Roots from Step 2

The roots \(x = -1\) and \(x = -\frac{10}{3}\) are already known from Inequality a. Use these to determine the intervals where the inequality is true.

Step 3: Test Intervals (Inequality b)

Using the intervals:1. \((-\infty, -\frac{10}{3})\)2. \(( -\frac{10}{3}, -1)\)3. \((-1, +\infty)\)Testing in these intervals, we found for:- \((-\infty, -\frac{10}{3})\), \(3(-4)^2 + 13(-4) + 10 > 0\)- \(( -\frac{10}{3}, -1)\), \(3(-2)^2 + 13(-2) + 10 \leq 0\)- \((-1, +\infty)\), \(3(0)^2 + 13(0) + 10 > 0\)Therefore, the solution is \((-\infty, -\frac{10}{3}) \cup (-1, +\infty)\).

Step 8: Graphical Representation

Graph both solutions. The inequality part \(a\) includes the line segment from \(-\frac{10}{3}\) to \(-1\) shaded, including the endpoints. Part \(b\) includes the areas extending infinitely left and right from these points, leaving \(-\frac{10}{3}\) to \(-1\) unshaded.

Key Concepts

Quadratic Equations

Quadratic equations are polynomial equations of degree two, typically in the form of \( ax^2 + bx + c = 0 \). They are called 'quadratic' because 'quad' means square, indicating the highest power is 2. These equations can have up to two solutions or roots, which can be real or complex numbers. By understanding these roots, we gain valuable insights into the behavior of the quadratic function.

A quadratic equation graphically represents a parabola. The roots of the equation are points where the parabola intersects the x-axis. Importantly, these roots give us clues about the solutions to inequalities involving the quadratic expression. When solving inequalities like \( 3x^2 + 13x + 10 \leq 0 \), identifying the roots helps us define critical intervals on the x-axis where the inequality changes its truth value.

Interval Testing

Interval testing is a useful method to solve quadratic inequalities by dividing the x-axis into segments based on the roots of the quadratic equation. Once roots are identified, they partition the number line into distinct intervals. For each interval, we choose a test point and evaluate the inequality to determine whether it holds true.

For example, with roots at \( x = -1 \) and \( x = -\frac{10}{3} \), the real number line divides into three intervals:

• \(( -\infty, -\frac{10}{3} )\)
• \( [-\frac{10}{3}, -1] \)
• \( (-1, +\infty) \)

Choosing a point in each interval determines if the inequality holds in that region. By testing, you find that the inequality \( 3x^2 + 13x + 10 \leq 0 \) is true in the interval \( [-\frac{10}{3}, -1] \). This method highlights the importance of intervals in deciding which sections of the number line satisfy the inequality conditions.

Graphical Representation

Graphing quadratic inequalities helps visualize where the solutions lie on the coordinate plane. For a quadratic inequality like \( 3x^2 + 13x + 10 \leq 0 \), imagine plotting the related quadratic equation \( 3x^2 + 13x + 10 = 0 \). The graph will show a parabola intersecting the x-axis at points corresponding to the roots \( x = -1 \) and \( x = -\frac{10}{3} \).

For inequality \( a \), shade the region between \( x = -\frac{10}{3} \) and \( x = -1 \) to indicate where the parabola is below or touches the x-axis since \( \leq \) includes the boundary. For inequality \( b \), highlight regions where the parabola is above the x-axis, and ensure this region is extended outward from the intersecting points, excluding the interval itself. Visualization through graphs provides a clear representation of the solutions, making it easier to comprehend which intervals satisfy the given inequalities. This visual aid is beneficial for enhancing understanding alongside analytical solutions.

Quadratic Formula

The quadratic formula is a powerful tool for finding the roots of any quadratic equation of the form \( ax^2 + bx + c = 0 \). By using the coefficients \( a \), \( b \), and \( c \), we calculate the roots using the formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]This formula stems from the process of "completing the square" and faithfully delivers the roots, whether they are real or complex.

In the context of the inequality \( 3x^2 + 13x + 10 \leq 0 \), substituting \( a = 3 \), \( b = 13 \), and \( c = 10 \) into the formula reveals the roots \( x = -1 \) and \( x = -\frac{10}{3} \). These roots serve as endpoints that split the x-axis into regions for evaluating the inequality. Mastery of the quadratic formula underpins analytical resolution of quadratic equations and inequalities, providing exact values necessary for precise interval definition and testing.