Table of contents

0. Review of Algebra4h 18m
1. Equations & Inequalities3h 18m
2. Graphs of Equations43m
3. Functions2h 17m
4. Polynomial Functions1h 44m
5. Rational Functions1h 23m
6. Exponential & Logarithmic Functions2h 28m
7. Systems of Equations & Matrices4h 6m
8. Conic Sections2h 23m
9. Sequences, Series, & Induction1h 22m
10. Combinatorics & Probability1h 45m

1. Equations & Inequalities

Linear Equations

College Algebra

1. Equations & Inequalities

Linear Equations

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1:52 minutes

Problem 63

Textbook Question

In Exercises 61–76, solve each absolute value equation or indicate that the equation has no solution.|x - 2| = 7

Verified step by step guidance

Understand that the absolute value equation |x - 2| = 7 can be split into two separate equations because the absolute value of a number is its distance from zero on the number line, which means it can be either positive or negative.

Set up the first equation by removing the absolute value and setting the expression inside equal to 7: x - 2 = 7.

Solve the first equation for x by adding 2 to both sides: x = 7 + 2.

Set up the second equation by removing the absolute value and setting the expression inside equal to -7: x - 2 = -7.

Solve the second equation for x by adding 2 to both sides: x = -7 + 2.

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Key Concepts

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

Absolute Value

Absolute value represents the distance of a number from zero on the number line, regardless of direction. It is denoted by vertical bars, such as |x|, and is always non-negative. For example, |3| = 3 and |-3| = 3. Understanding absolute value is crucial for solving equations that involve it, as it leads to two possible cases.

Solving Absolute Value Equations

To solve an absolute value equation like |x - 2| = 7, you must consider the two scenarios that arise from the definition of absolute value. This means setting up two separate equations: x - 2 = 7 and x - 2 = -7. Solving these equations will yield the potential solutions for x, which must then be verified in the original equation.

Verification of Solutions

After finding potential solutions from an absolute value equation, it is essential to verify each solution by substituting it back into the original equation. This step ensures that the solutions satisfy the equation, as sometimes extraneous solutions can arise during the solving process. Verification confirms the validity of the solutions found.