Ch. 1 - Equations and Inequalities

Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook

Blitzer 8th Edition

Ch. 1 - Equations and Inequalities

Problem 94

Chapter 2, Problem 94

Solve the equations containing absolute value in Exercises 94–95. |2x+1| = 7

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Recognize that the equation |2x + 1| = 7 involves an absolute value. The absolute value equation |A| = B can be rewritten as two separate equations: A = B and A = -B, provided B ≥ 0.

Rewrite the given equation |2x + 1| = 7 as two separate equations: 2x + 1 = 7 and 2x + 1 = -7.

Solve the first equation, 2x + 1 = 7, by isolating x. Subtract 1 from both sides to get 2x = 6, then divide both sides by 2 to find x.

Solve the second equation, 2x + 1 = -7, by isolating x. Subtract 1 from both sides to get 2x = -8, then divide both sides by 2 to find x.

Combine the solutions from both equations to express the final solution set for x.

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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|. For example, |3| equals 3, and |-3| also equals 3. Understanding absolute value is crucial for solving equations that involve it, as it leads to two possible cases based on the definition.

Solving Absolute Value Equations

To solve an equation involving absolute value, you must consider both the positive and negative scenarios. For the equation |2x + 1| = 7, you set up two separate equations: 2x + 1 = 7 and 2x + 1 = -7. Solving these equations will yield the possible values of x that satisfy the original equation.

Checking Solutions

After finding potential solutions for an absolute value equation, it is essential to check each solution by substituting it back into the original equation. This step ensures that the solutions are valid and satisfy the equation, as sometimes extraneous solutions can arise during the solving process.

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In Exercises 95–102, use interval notation to represent all values of x satisfying the given conditions. $y_1 = $\frac{x}{2}$ + 3, $\quad$ y_2 = $\frac{x}{3}$ + $\frac{5}{2}$, $\quad$ $\text{and}$ $\quad$ y_1 $\leq$ y_2$

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