Ch. 1 - Equations and Inequalities

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

Lial 13th Edition

Ch. 1 - Equations and Inequalities

Problem 3

Chapter 2, Problem 3

Match each equation or inequality in Column I with the graph of its solution set in Column II. | x | > -7

Verified step by step guidance

Understand the inequality given: \(|x| > -7\). The absolute value \(|x|\) represents the distance of \(x\) from zero on the number line, which is always greater than or equal to zero.

Recall that absolute value expressions are always non-negative, so \(|x| \geq 0\) for all real numbers \(x\). This means \(|x|\) can never be less than zero, and certainly never less than a negative number like \(-7\).

Since \(|x|\) is always greater than or equal to zero, and zero is greater than any negative number, the inequality \(|x| > -7\) is true for every real number \(x\).

Therefore, the solution set to the inequality \(|x| > -7\) is all real numbers, which means the graph representing this solution will be the entire number line.

When matching this inequality to a graph, look for the graph that shows all real numbers shaded or included, indicating the solution set is all real numbers.

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Key Concepts

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

Absolute Value Inequalities

Absolute value inequalities involve expressions with absolute value symbols, such as |x| > a. They describe the distance of a number from zero on the number line. Understanding how to interpret and solve these inequalities is essential for matching them to their solution sets.

Properties of Absolute Value

The absolute value of any real number is always non-negative. This means |x| ≥ 0 for all x. Recognizing that |x| > -7 is always true because absolute values cannot be negative helps in determining the solution set.

Graphing Solution Sets on the Number Line

Graphing solution sets involves shading regions on the number line that satisfy the inequality. For absolute value inequalities, this often results in two intervals or the entire number line, depending on the inequality's conditions.

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Match the inequality in each exercise in Column I with its equivalent interval notation in Column II. -2 < x ≤ 6

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Match the equation in Column I with its solution(s) in Column II. x² - 5 = 0

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Match each equation or inequality in Column I with the graph of its solution set in Column II. | x | > 7

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Textbook Question

Answer each question. Sides of a Right TriangleTo solve for the lengths of the right triangle sides, which equation is correct?

A. x^2=(2x-2)^2+(x+4)^2 B. x^2+(x+4)^2=(2x-2)^2 C. x^2=(2x-2)^2-(x+4)^2 D. x^2+(2x-2)^2=(x+4)^2