#### Chapter 2 Equations in one Variable

Section 2.2 Absolute Value Equations

# 2.2.2 Carry out a Case Analysis

##### Info 2.2.2

To solve an absolute value equation two cases are distinguished:
• For all values of $x$ for which the absolute value term is non-negative the absolute value can be omitted or replaced by simple brackets, respectively.
• For all values of $x$ for which the absolute value term is negative the term is bracketed and negated.
Then, the solution sets from the case analyses will be restricted to satisfy the case conditions. Only if this procedure is finished for all cases, the solution subsets will be merged to the solution set of the initial equation.

For solving absolute value equations it is important to write down the solution steps correctly and to distinguish the cases clearly.
The following video demonstrates a detailed written solution of the absolute value equation $|2x-4|=6$ by case analysis.
Video 1: Carry out a case analysis.

The case analysis presented in the video can be written briefly as follows:

$|2x-4|\mathrm{ }=\mathrm{ }\left\{\begin{array}{cc}\hfill 2x-4& \text{for}\mathrm{ }x\ge 2\hfill \\ \hfill -2x+4& \text{for}\mathrm{ }x<2\hfill \end{array}\mathrm{ }=\mathrm{ }\left\{\begin{array}{cc}\hfill 2x-4& \text{for}\mathrm{ }x\ge 2\hfill \\ \hfill -2x+4& \text{otherwise}\hfill \end{array} .$

In an input field you would type this as falls(x>=2;2*x-4;-2*x+4). Or alternatively as falls(x<2;-2*x+4;2*x-4).
##### Exercise 2.2.3
Describe the values of the expression $2·|x-4|$ by a case analysis:
$2·|x-4|$ =
.
Enter the case analysis in the form falls(CONDITION;W1;W2), where W1 is the value of the expression if the corresponding condition is satisfied. Do not use the absolute value function.

##### Exercise 2.2.4
Reproduce the steps shown in the video 2.2.1 to solve the absolute value equation $|6+3x|=12$.
The case analysis reads shortly $|6+3x|$ =
.
Enter the case analysis in the form falls(CONDITION;W1;W2), where W1 is the value of the expression if the corresponding condition is satisfied. You can copy one of the input examples into the input field and adapt it to the new equation.
Finding the solution for each case and checking the case conditions leads to the solution set $L$$=$
for the equation $|6+3x|=12$.

Mengen können in der Form $\left\{$a;b;c;...$\right\}$ eingegeben werden.

You can practise the stepwise solution of absolute value equations within the following exercise.
##### Exercise 2.2.5
Lösen Sie die Gleichung $2|6x+3|+4x-4=5|-3x-3|-1$