Problem 25 - The Calculus with Analytic Geometry

Question

In Exercises 11 through 32 , find the solution set of the given inequality and illustrate the solution on the real number $$ 1-x-2 x^{2} \geq 0 $$

Step-by-Step Solution

Answer

The solution set is x ∈ (-∞, -1/2] ∪ [1, ∞).

Step 1: Rewrite the Inequality

Start by rewriting the inequality in a standard quadratic form. Given: 1 - x - 2x^2 Move all terms to one side of the inequality. We have: -2x^2 - x + 1 This can be rewritten as: -2x^2 - x + 1

Step 2: Factor the Quadratic

To solve the inequality, factor the quadratic expression if possible. It factors as: (2x + 1)(x - 1)

Step 3: Find Critical Points

These factors give us the critical points where the expression equals zero. Solve (2x + 1)(x - 1) = 0 to find the critical points: 2x + 1 = 0 x = -1/2 x - 1 = 0 x = 1 So the critical points are x = -1/2 and x = 1.

Step 4: Test Intervals Between Critical Points

Check the sign of the quadratic expression between the intervals determined by the critical points. Choose test points in the intervals: Interval 1: (-∞, -1/2), test point x = -1 Interval 2: (-1/2, 1), test point x = 0 Interval 3: (1, ∞), test point x = 2 Check the sign of (2x + 1)(x - 1) in these intervals.

Step 5: Determine the Solution Set

Evaluate the signs using the test points: In Interval 1 (-∞, -1/2), (2(-1) + 1)((-1) - 1) > 0 In Interval 2 (-1/2, 1), (2(0) + 1)((0) - 1) < 0 In Interval 3 (1, ∞), (2(2) + 1)((2) - 1) > 0 The solution sets are the intervals where the expression is non-negative. So, the solution set is x ∈ (-∞, -1/2] ∪ [1, ∞).

Step 6: Illustrate the Solution on a Number Line

Illustrate the solution set on a number line, marking x = -1/2 and x = 1, and shading the intervals (-∞, -1/2] and [1, ∞).

Key Concepts

Quadratic Inequalities

Quadratic inequalities are inequalities that involve quadratic expressions. These inequalities come in forms like ax^2 + bx + c < 0, ax^2 + bx + c > 0, and their non-strict counterparts. The goal is to find the range of x values that satisfy the inequality. To solve these, we often transform the quadratic into standard form and find key points where the expression equals zero.

Factoring Quadratics

Factoring quadratics is crucial in solving quadratic inequalities. Factoring breaks down the quadratic expression into simpler expressions (factors). For example, the expression 1 - x - 2x^2 can be rewritten as -2x^2 - x + 1. Factoring simplifies this to (2x + 1)(x - 1). These factors help identify the critical points by setting each factor to zero.

Critical Points

Critical points are the x-values where the quadratic expression equals zero. They are found by solving the factored equation. For our example, we solve (2x + 1)(x - 1) = 0 to get x = -1/2 and x = 1. These points are crucial as they divide the number line into intervals to test the signs of the expression.

Interval Testing

Interval testing involves checking the sign of the quadratic expression within the regions defined by the critical points. By choosing test points within these intervals, we can determine where the expression is positive or negative. For instance, for the intervals (-∞, -1/2], (-1/2, 1), and [1, ∞), we select points like -1, 0, and 2 and substitute them into (2x + 1)(x - 1). This determines if the expression meets the inequality condition.

Number Line Illustration

Number line illustration visually represents the solution set of the inequality. Draw the number line, mark the critical points found (x = -1/2 and x = 1), and shade the intervals that satisfy the quadratic inequality. For our problem, shade (-∞, -1/2] and [1, ∞). This graphical approach helps in understanding the solution set clearly.