Ch. 1 - Equations and Inequalities

Lial13th EditionCollege AlgebraISBN: 9780136881063Not the one you use?Change textbook

Lial 13th Edition

Ch. 1 - Equations and Inequalities

Problem 92

Chapter 2, Problem 92

Solve each inequality. Give the solution set using interval notation. x²-3x ≥ 5

Verified step by step guidance

Start by rewriting the inequality to have zero on one side: subtract 5 from both sides to get $x^{2} - 3x - 5 \geq 0$.

Next, find the roots of the quadratic equation $x^{2} - 3x - 5 = 0$ by using the quadratic formula: $x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}$, where $a=1$, $b=-3$, and $c=-5$.

Calculate the discriminant $\Delta = b^{2} - 4ac = (-3)^{2} - 4(1)(-5)$ to determine the nature of the roots.

Use the roots found to divide the number line into intervals. Test a value from each interval in the inequality $x^{2} - 3x - 5 \geq 0$ to determine where the inequality holds true.

Express the solution set in interval notation based on the intervals where the inequality is satisfied, including the points where the expression equals zero.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.

Video duration:

10m

Was this helpful?

Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Solving Quadratic Inequalities

Solving quadratic inequalities involves finding the values of the variable that make the inequality true. This typically requires rewriting the inequality in standard form, factoring or using the quadratic formula to find critical points, and then testing intervals to determine where the inequality holds.

Interval Notation

Interval notation is a way to represent sets of numbers on the number line. It uses parentheses () for values not included and brackets [] for values included, describing continuous ranges such as (a, b), [a, b), or (-∞, c]. This notation concisely expresses solution sets of inequalities.

Sign Analysis of Quadratic Expressions

Sign analysis involves determining where a quadratic expression is positive, negative, or zero by examining its roots and the shape of its graph. Since a quadratic opens upward or downward, the sign of the expression changes at its roots, helping identify solution intervals for inequalities.

Recommended video:

06:36

Solving Quadratic Equations Using The Quadratic Formula

Related Practice

Textbook Question

Solve each equation. (2x-1)^2/3+2(2x-1)^1/3-3=0

Textbook Question

Simplify each power of i. i²⁷

Textbook Question

Simplify each power of i. i²³

Textbook Question

Evaluate the discriminant for each equation. Then use it to determine the number of distinct solutions, and tell whether they are rational, irrational, or nonreal complex numbers. (Do not solve the equation.) See Example 9.

$8 x^{2} - 72 = 0$

Textbook Question

Use the method described in Exercises 83–86, if applicable, and properties of absolute value to solve each equation or inequality. (Hint: Exercises 99 and 100 can be solved by inspection.) | x² - 9 | = x + 3

views

Textbook Question

Simplify each power of i. i²⁶

views