Problem 42 - Precalculus Mathematics for Calculus

Question

Solve the nonlinear inequality. Express the solution using interval notation and graph the solution set. $$x^{2}

Step-by-Step Solution

Answer

The solution is \((-1, 2)\).

Step 1: Rearrange the Inequality

First, rearrange the inequality to bring all terms to one side. The original inequality is \(x^2 < x + 2\). Move \(x\) and \(2\) to the left side:\[x^2 - x - 2 < 0\]

Step 2: Solve the Corresponding Equation

To solve the inequality, first solve the related equation \(x^2 - x - 2 = 0\).Factoring the quadratic:\[(x-2)(x+1) = 0\]The solutions to the equation are \(x = 2\) and \(x = -1\).

Step 3: Test Intervals Based on Critical Points

The critical points found are \(x = -1\) and \(x = 2\), dividing the real number line into three intervals:1. \((-\infty, -1)\)2. \((-1, 2)\)3. \((2, \infty)\)Choose a test point from each interval to determine where \(x^2 - x - 2 < 0\).

Step 4: Evaluate Test Points

Evaluate \(x^2 - x - 2\) at points from each interval:- For \(x = -2\) from \((-\infty, -1)\): \\((-2)^2 - (-2) - 2 = 4 + 2 - 2 = 4\) (positive)- For \(x = 0\) from \((-1, 2)\): \\(0^2 - 0 - 2 = -2\) (negative)- For \(x = 3\) from \((2, \infty)\): \\(3^2 - 3 - 2 = 9 - 3 - 2 = 4\) (positive)The inequality is satisfied in the interval \((-1, 2)\).

Step 5: Express the Solution in Interval Notation

Since \(x^2 - x - 2 < 0\) is satisfied in the interval \((-1, 2)\), and we are dealing with a strict inequality (\(<\)), we do not include the endpoints. Therefore, the solution in interval notation is \((-1, 2)\).

Key Concepts

Quadratic Inequalities

Quadratic inequalities involve expressions with a variable raised to the power of two. These are inequalities like \(x^2 < x + 2\) which can be rearranged to the standard quadratic form: \(ax^2 + bx + c < 0\) or \(ax^2 + bx + c > 0\). A quadratic inequality shows a relationship of greater than or less than between two expressions, rather than an equality as in the case of quadratic equations.
Solving a quadratic inequality typically involves the following steps:

• Rearrange the inequality so that all terms are on one side and zero is on the other side.
• Solve the related quadratic equation to find the critical points.
• Use these critical points to divide the number line into intervals.
• Test each interval to determine where the inequality holds true.
• Express the solution using interval notation.

Understanding these steps helps in effectively handling any quadratic inequality problem.

Interval Notation

Interval notation is a concise way of expressing sets of numbers, especially suited for defining the solution sets of inequalities. In mathematics, it allows us to denote intervals of real numbers from one endpoint to another.
When expressed in interval notation, brackets and parentheses are used:

• Closed Interval [a, b]: Both endpoints, \(a\) and \(b\), are included in the interval.
• Open Interval (a, b): Neither \(a\) nor \(b\) are included.
• Half-Open Interval [a, b) or (a, b]: One endpoint is included, and the other is not.

For quadratic inequalities, after determining where the inequality holds, the solution is often expressed as an interval where the inequality is satisfied.
This allows a quick representation of potentially infinite sets of solutions in a finite form.

Critical Points

Critical points in the context of quadratic inequalities are key values which help to determine the boundaries or transition points for solutions to the inequality. These are the solutions to the quadratic equation derived from the inequality.
For example, consider the inequality \(x^2 - x - 2 < 0\). Solving the related equation \(x^2 - x - 2 = 0\) gives critical points. Factoring, we solve:\[(x - 2)(x + 1) = 0\]which results in the critical points \(x = 2\) and \(x = -1\).
These points divide the number line into intervals which can be evaluated separately to determine where the inequality is true. Testing regions divided by these critical points helps you decide where the polynomial is either positive or negative, ultimately letting you find where the inequality is satisfied.
Understanding critical points is crucial, as it forms the basis of partitioning the number line and finding the solution set efficiently.