Question

Solve each equation. $\left|\frac{1}{5} x+2\right|-8=-8$

   Solve each equation.
$\left|\frac{1}{5} x+2\right|-8=-8$

Elementary and Intermediate Algebra

Elementary and Intermediate Algebra

Alan S. Tussy, R.… 5th Edition

Chapter 8, Problem 39

Instant Answer

Solved by Expert Cory Kuzinski

04/11/2020

Step 1

We can do this by adding 8 to both sides of the equation: $$\left|\frac{1}{5} x+2\right|-8+8=-8+8$$ This simplifies to: $$\left|\frac{1}{5} x+2\right|=0$$ Show more…

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Solve each equation. $\left|\frac{1}{5} x+2\right|-8=-8$

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Key Concepts

Absolute Value Function

The absolute value of a number or expression represents its distance from zero on the number line, meaning it is always non-negative. This concept is fundamental when dealing with absolute value equations, as it directly affects the possible solutions since the expression inside the absolute value can be positive or negative, but the output is always non-negative.

Solving Absolute Value Equations

When solving equations that involve absolute value expressions, the first step is often to isolate the absolute value on one side of the equation. Once isolated, one considers the nature of absolute values, knowing that the absolute value output is always non-negative. This implies that setting the absolute value equal to a non-negative number provides one or two cases to solve, whereas setting it equal to a negative number results in no solutions.

Isolating the Absolute Value Expression

In equations containing absolute values, it is crucial to isolate the absolute value term before proceeding to the next solving step. This isolation simplifies the equation and makes explicit the condition that the absolute value is equal to a certain non-negative value, which aids in setting up the proper case or condition(s) to solve for the variable.

Setting the Absolute Value Equal to Zero

When an absolute value expression is set equal to zero, it implies that the expression inside the absolute value must itself be zero, as absolute value can only yield zero if the input is zero. This simplifies the problem to a single linear or non-linear equation when determining the solution.

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