Question

Solve each absolute value equation. $|7 z|+1=22$

   Solve each absolute value equation.
$|7 z|+1=22$

Beginning and Intermediate Algebra

Beginning and Intermediate Algebra

Elayn Martin-Gay 4th Edition

Chapter 9, Problem 36

Instant Answer

Solved by Expert Ali Soave

03/15/2020

Step 1

Step 1: Subtract 1 from both sides of the equation to isolate the absolute value term: \[|7z| = 22 - 1\] \[|7z| = 21\] Show more…

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Solve each absolute value equation. $|7 z|+1=22$

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Key Concepts

Absolute Value Function

The absolute value function measures the distance of a number from zero on the number line, regardless of its sign. In the context of equations, it ensures the expression inside the absolute value is treated as a nonnegative quantity, which is important when setting up solutions.

Isolating the Absolute Value Expression

A key strategy in solving absolute value equations is to first isolate the absolute value on one side of the equation. This simplifies the equation and makes it possible to apply the definition of absolute value to set up further case analyses.

Two-Case Analysis of Absolute Value Equations

After isolating the absolute value, the equation is split into two cases: one where the inside of the absolute value is equal to the given positive value, and another where it is equal to the negative of that value. This approach relies on the property that if |A| = B (with B being nonnegative), then either A = B or A = -B.

Solving Linear Equations

Once the absolute value equation is split into two separate equations, each resulting equation is typically a linear equation that can be solved using standard algebraic techniques. This step involves isolating the variable and performing arithmetic operations to find its value.

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