Ch. 1 - Equations and Inequalities

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

Lial 13th Edition

Ch. 1 - Equations and Inequalities

Problem 34

Chapter 2, Problem 34

Solve each inequality. Give the solution set in interval notation. 5| x + 1 | > 12

Verified step by step guidance

Start by isolating the absolute value expression. Divide both sides of the inequality $5| x + 1 | > 10$ by 5 to get $| x + 1 | > 2$.

Recall that the inequality $| A | > B$ (where $B > 0$) means that $A > B$ or $A < -B$. Apply this to $| x + 1 | > 2$ to write two inequalities: $x + 1 > 2$ or $x + 1 < -2$.

Solve each inequality separately. For $x + 1 > 2$, subtract 1 from both sides to get $x > 1$. For $x + 1 < -2$, subtract 1 from both sides to get $x < -3$.

Combine the two solution sets. The solution to the original inequality is all $x$ such that $x > 1$ or $x < -3$.

Express the solution set in interval notation as $(-\infty, -3) \cup (1, \infty)$.

Verified video answer for a similar problem:

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

Video duration:

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 where the absolute value of a variable or expression is compared to a number. To solve them, consider the definition of absolute value as distance from zero, leading to two cases: one where the expression inside is greater than the positive number, and one where it is less than the negative of that number.

Isolating the Absolute Value Expression

Before solving an absolute value inequality, isolate the absolute value term on one side of the inequality. This often involves dividing or multiplying both sides by a positive number, which does not change the inequality direction, making the inequality easier to analyze and solve.

Interval Notation

Interval notation is a way to represent solution sets of inequalities using intervals on the number line. It uses parentheses for open intervals (excluding endpoints) and brackets for closed intervals (including endpoints), providing a concise and clear way to express all values that satisfy the inequality.

Recommended video:

05:18

Interval Notation

Related Practice

Textbook Question

Solve each equation. (x+4)/2x = (x-1)/3

views

Textbook Question

Radius of a Can A can of Blue Runner Red Kidney Beans has surface area 371 cm². Its height is 12 cm. What is the radius of the circular top? Round to the nearest hundredth.

Textbook Question

Solve each problem. See Example 4. Cody sells some property for \$240,000. The money will be paid off in two ways: a short-term note at 2% interest and a long-term note at 2.5%. Find the amount of each note if the total annual interest paid is \$5500.

views

Textbook Question

Solve each equation using the square root property. (x - 4)² = -5

views

Textbook Question

Find each product or quotient. Simplify the answers. √-30 / √-10

Textbook Question

Solve each inequality. Give the solution set in interval notation. 2>-6x+3>-3