Problem 14 - Calculus Early Transcendentals: Pearson New International Edition

Question

Express the solution set of the given inequality in interval notation and sketch its graph. $$ 4 x^{2}-5 x-6<0 $$

Step-by-Step Solution

Answer

The solution set is \((-\frac{3}{4}, 2)\).

Step 1: Identify the Quadratic Equation

The given inequality is \( 4x^2 - 5x - 6 < 0 \). We start by solving the related quadratic equation to find the critical points. The related equation is \( 4x^2 - 5x - 6 = 0 \).

Step 2: Factor the Quadratic Equation

To factor \( 4x^2 - 5x - 6 = 0 \), we look for two numbers whose product is \(-24\) (4 multiplied by -6) and whose sum is \(-5\). These numbers are 3 and -8, so we rewrite the equation as \((4x^2 + 3x) - (8x + 6) = 0\).

Step 3: Simplify the Factored Expression

Factor by grouping: \((4x^2 + 3x) - (8x + 6) = x(4x + 3) - 2(4x + 3) = (4x + 3)(x - 2) = 0\). This gives us the critical points \(x = -\frac{3}{4}\) and \(x = 2\).

Step 4: Determine the Solution Set

Plot the critical points on a number line: \(-\frac{3}{4}\) and \(2\). Test intervals between critical points to determine where the inequality is satisfied. Test points: for \(x < -\frac{3}{4}, x = -1\); for \(-\frac{3}{4} < x < 2, x = 0\); for \(x > 2, x = 3\).

Step 5: Test Interval Solutions

Substitute test points into the inequality \(4x^2 - 5x - 6 < 0 \):- For \(x = -1\), \(4(-1)^2 - 5(-1) - 6 = -7\) (false).- For \(x = 0\), \(-6 < 0\) (true).- For \(x = 3\), \(36 - 15 - 6 = 15\) (false). Thus, the solution where the inequality holds is the interval \(-\frac{3}{4} < x < 2\).

Step 6: Express in Interval Notation

The solution set for the inequality is \((-\frac{3}{4}, 2)\) based on the intervals tested.

Step 7: Sketch the Graph

Plot \(y = 4x^2 - 5x - 6\) with critical points at \(x = -\frac{3}{4}\) and \(x = 2\). Shade the region between these points on the x-axis that satisfy \(4x^2 - 5x - 6 < 0\). This is the part of the parabola below the x-axis.

Key Concepts

Quadratic Inequality

A quadratic inequality is similar to a quadratic equation but instead of an equality symbol, it uses inequality symbols like less than <, greater than >, less than or equal to ≤, and greater than or equal to ≥. These inequalities generally define a range of values that satisfy the inequality condition for a given quadratic expression. Quadratic inequalities can describe intervals on the real number line that correspond to where the output, usually represented by a parabola, lies above or below the x-axis.
When solving a quadratic inequality such as \(4x^2 - 5x - 6 < 0\), the first step is to find the roots of the corresponding quadratic equation \(4x^2 - 5x - 6 = 0\). These roots, or critical points, will help determine where the inequality is satisfied on the number line. The solution of the inequality is the set of values for which the quadratic expression evaluates to less than zero and is often expressed in interval notation once verified by plotting or solving valences.
Understanding quadratic inequalities provides valuable insight into regions of the number line that meet specific criteria set by the inequality. It's necessary to carefully analyze solutions to account for all intervals and endpoints correctly.

Factoring Quadratic Equations

Factoring is a crucial skill for solving quadratic equations of the form \(ax^2 + bx + c = 0\). An effective factorization breaks down the equation into simpler binomial expressions that can be solved to find the critical points. For the equation \(4x^2 - 5x - 6 = 0\), the goal is to express it in a factored form: \((4x + 3)(x - 2) = 0\).
To factor quadratic equations, begin by identifying two numbers that multiply to give the product of the coefficient of \(x^2\) (a) and the constant term (c), while also adding up to the middle coefficient (b). Here, the product is \(-24\) and the sum is \(-5\). The numbers \(3\) and \(-8\) fit, allowing the middle term to be rewritten to facilitate grouping.

• Group the terms: \((4x^2 + 3x) - (8x + 6)\)
• Factor by grouping: \(x(4x + 3) - 2(4x + 3)\)
• Factor out the common binomial: \((4x + 3)(x - 2)\)

Factoring allows us to solve each binomial equation, finding the critical points or roots, which are crucial in analyzing the entire inequality.

Number Line Analysis

Number line analysis is a vital tool in solving inequalities. This process involves checking where the inequality holds true by partitioning the number line according to the critical points found from the factored quadratic equation. For \((4x + 3)(x - 2) < 0\), the critical points are found at \(x = -\frac{3}{4}\) and \(x = 2\).
Number line analysis requires choosing test points in the intervals formed by these critical points to determine where the original inequality \(4x^2 - 5x - 6 < 0\) is satisfied. Here are the steps:

• Mark the critical points on the number line.
• Check intervals:\(\text{Interval 1: } x < -\frac{3}{4} \), e.g., \(x = -1\) fails.
• \(\text{Interval 2: } -\frac{3}{4} < x < 2\), e.g., \(x = 0\) passes.
• \(\text{Interval 3: } x > 2\), e.g., \(x = 3\) fails.

The interval where the quadratic inequality holds is between \(-\frac{3}{4}\) and \(2\), written as \((-\frac{3}{4}, 2)\), excluding the endpoints as they are not part of the solution.

Graphing Inequalities

Graphing inequalities is a powerful visual tool to understand solutions for equations and inequalities. When graphing the quadratic inequality \(4x^2 - 5x - 6 < 0\), the graph of the corresponding quadratic expression \(y = 4x^2 - 5x - 6\) is a parabola that opens upwards. The roots or critical points \(x = -\frac{3}{4}\) and \(x = 2\) allow us to observe where the parabola dips below the x-axis.
For a visual representation:

• Draw the x-axis and y-axis.
• Sketch the parabola based on the quadratic function, noting it opens upwards with roots at the calculated critical points.
• Shade the region between \(x = -\frac{3}{4}\) and \(x = 2\) as this is where \(y < 0\).

This shaded section visually confirms the intervals on the number line analysis and provides an intuitive understanding of inequality solutions. It clearly shows which sections of the parabola satisfy the given inequality condition. The graph along with the interval notation \((-\frac{3}{4}, 2)\) collectively represent the solution set.