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Solution - Absolute value equations

Exact form:

x = 8, 2

x=8 , 2

See steps

Step-by-step explanation

1. Rewrite the equation without absolute value bars

Use the rules:
$| x | = | y |$ → $x = \pm y$ and $| x | = | y |$ → $\pm x = y$
to write all four options of the equation
$| 1.6 x - 5 | = | x - 0.2 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| 1.6 x - 5 \| = \| x - 0.2 \|$
$x = + y$	$(1.6 x - 5) = (x - 0.2)$
$x = - y$	$(1.6 x - 5) = - (x - 0.2)$
$+ x = y$	$(1.6 x - 5) = (x - 0.2)$
$- x = y$	$- (1.6 x - 5) = (x - 0.2)$

When simplified, equations $x = + y$ and $+ x = y$ are the same and equations $x = - y$ and $- x = y$ are the same, so we end up with only 2 equations:

$\| x \| = \| y \|$	$\| 1.6 x - 5 \| = \| x - 0.2 \|$
$x = + y$ , $+ x = y$	$(1.6 x - 5) = (x - 0.2)$
$x = - y$ , $- x = y$	$(1.6 x - 5) = - (x - 0.2)$

2. Solve the two equations for x

10 additional steps

$(1.6 x - 5) = (x - 0.2)$

Subtract from both sides:

$(1.6 x - 5) - x = (x - 0.2) - x$

Group like terms:

$(1.6 x - x) - 5 = (x - 0.2) - x$

Simplify the arithmetic:

$0.6 x - 5 = (x - 0.2) - x$

Group like terms:

$0.6 x - 5 = (x - x) - 0.2$

Simplify the arithmetic:

$0.6 x - 5 = - 0.2$

Add to both sides:

$(0.6 x - 5) + 5 = - 0.2 + 5$

Simplify the arithmetic:

$0.6 x = - 0.2 + 5$

Simplify the arithmetic:

$0.6 x = 4.8$

Divide both sides by :

$\frac{(0.6 x)}{0.6} = \frac{4.8}{0.6}$

Simplify the arithmetic:

$x = \frac{4.8}{0.6}$

Simplify the arithmetic:

$x = 8$

11 additional steps

$(1.6 x - 5) = - (x - 0.2)$

Expand the parentheses:

$(1.6 x - 5) = - x + 0.2$

Add to both sides:

$(1.6 x - 5) + x = (- x + 0.2) + x$

Group like terms:

$(1.6 x + x) - 5 = (- x + 0.2) + x$

Simplify the arithmetic:

$2.6 x - 5 = (- x + 0.2) + x$

Group like terms:

$2.6 x - 5 = (- x + x) + 0.2$

Simplify the arithmetic:

$2.6 x - 5 = 0.2$

Add to both sides:

$(2.6 x - 5) + 5 = 0.2 + 5$

Simplify the arithmetic:

$2.6 x = 0.2 + 5$

Simplify the arithmetic:

$2.6 x = 5.2$

Divide both sides by :

$\frac{(2.6 x)}{2.6} = \frac{5.2}{2.6}$

Simplify the arithmetic:

$x = \frac{5.2}{2.6}$

Simplify the arithmetic:

$x = 2$

3. List the solutions

$x = 8, 2$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | 1.6 x - 5 |$
$y = | x - 0.2 |$
The equation is true where the two lines cross.

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Why learn this

We encounter absolute values almost daily. For example: If you walk 3 miles to school, do you also walk minus 3 miles when you go back home? The answer is no because distances use absolute value. The absolute value of the distance between home and school is 3 miles, there or back.
In short, absolute values help us deal with concepts like distance, ranges of possible values, and deviation from a set value.

Solution - Absolute value equations

Step-by-step explanation

1. Rewrite the equation without absolute value bars

2. Solve the two equations for x

3. List the solutions

4. Graph

Why learn this

Terms and topics

Related links

Solution - Absolute value equations

Step-by-step explanation

1. Rewrite the equation without absolute value bars

2. Solve the two equations for x

3. List the solutions

4. Graph

Why learn this

Terms and topics

Related links

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