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Solution - Absolute value equations

Exact form:

x = \frac{7}{3}, \frac{9}{5}

x=\frac{7}{3} , \frac{9}{5}

Mixed number form:

x = 2 \frac{1}{3}, 1 \frac{4}{5}

x=2\frac{1}{3} , 1\frac{4}{5}

Decimal form:

x = 2.333, 1.8

x=2.333 , 1.8

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
$| x - 1 | = 4 | x - 2 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| x - 1 \| = 4 \| x - 2 \|$
$x = + y$	$(x - 1) = 4 (x - 2)$
$x = - y$	$(x - 1) = 4 (- (x - 2))$
$+ x = y$	$(x - 1) = 4 (x - 2)$
$- x = y$	$- (x - 1) = 4 (x - 2)$

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:

$\| x \| = \| y \|$	$\| x - 1 \| = 4 \| x - 2 \|$
$x = + y$ , $+ x = y$	$(x - 1) = 4 (x - 2)$
$x = - y$ , $- x = y$	$(x - 1) = 4 (- (x - 2))$

2. Solve the two equations for x

13 additional steps

$(x - 1) = 4 \cdot (x - 2)$

Expand the parentheses:

$(x - 1) = 4 x + 4 \cdot - 2$

Simplify the arithmetic:

$(x - 1) = 4 x - 8$

Subtract from both sides:

$(x - 1) - 4 x = (4 x - 8) - 4 x$

Group like terms:

$(x - 4 x) - 1 = (4 x - 8) - 4 x$

Simplify the arithmetic:

$- 3 x - 1 = (4 x - 8) - 4 x$

Group like terms:

$- 3 x - 1 = (4 x - 4 x) - 8$

Simplify the arithmetic:

$- 3 x - 1 = - 8$

Add to both sides:

$(- 3 x - 1) + 1 = - 8 + 1$

Simplify the arithmetic:

$- 3 x = - 8 + 1$

Simplify the arithmetic:

$- 3 x = - 7$

Divide both sides by :

$\frac{(- 3 x)}{- 3} = \frac{- 7}{- 3}$

Cancel out the negatives:

$\frac{3 x}{3} = \frac{- 7}{- 3}$

Simplify the fraction:

$x = \frac{- 7}{- 3}$

Cancel out the negatives:

$x = \frac{7}{3}$

14 additional steps

$(x - 1) = 4 \cdot (- (x - 2))$

Expand the parentheses:

$(x - 1) = 4 \cdot (- x + 2)$

$(x - 1) = 4 \cdot - x + 4 \cdot 2$

Group like terms:

$(x - 1) = (4 \cdot - 1) x + 4 \cdot 2$

Multiply the coefficients:

$(x - 1) = - 4 x + 4 \cdot 2$

Simplify the arithmetic:

$(x - 1) = - 4 x + 8$

Add to both sides:

$(x - 1) + 4 x = (- 4 x + 8) + 4 x$

Group like terms:

$(x + 4 x) - 1 = (- 4 x + 8) + 4 x$

Simplify the arithmetic:

$5 x - 1 = (- 4 x + 8) + 4 x$

Group like terms:

$5 x - 1 = (- 4 x + 4 x) + 8$

Simplify the arithmetic:

$5 x - 1 = 8$

Add to both sides:

$(5 x - 1) + 1 = 8 + 1$

Simplify the arithmetic:

$5 x = 8 + 1$

Simplify the arithmetic:

$5 x = 9$

Divide both sides by :

$\frac{(5 x)}{5} = \frac{9}{5}$

Simplify the fraction:

$x = \frac{9}{5}$

3. List the solutions

$x = \frac{7}{3}, \frac{9}{5}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | x - 1 |$
$y = 4 | x - 2 |$
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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