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Solution - Absolute value equations

Exact form:

x = 2, \frac{4}{9}

x=2 , \frac{4}{9}

Decimal form:

x = 2, 0.444

x=2 , 0.444

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
$| 4 x - 1 | = | 5 x - 3 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| 4 x - 1 \| = \| 5 x - 3 \|$
$x = + y$	$(4 x - 1) = (5 x - 3)$
$x = - y$	$(4 x - 1) = - (5 x - 3)$
$+ x = y$	$(4 x - 1) = (5 x - 3)$
$- x = y$	$- (4 x - 1) = (5 x - 3)$

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:

$\| x \| = \| y \|$	$\| 4 x - 1 \| = \| 5 x - 3 \|$
$x = + y$ , $+ x = y$	$(4 x - 1) = (5 x - 3)$
$x = - y$ , $- x = y$	$(4 x - 1) = - (5 x - 3)$

2. Solve the two equations for x

10 additional steps

$(4 x - 1) = (5 x - 3)$

Subtract from both sides:

$(4 x - 1) - 5 x = (5 x - 3) - 5 x$

Group like terms:

$(4 x - 5 x) - 1 = (5 x - 3) - 5 x$

Simplify the arithmetic:

$- x - 1 = (5 x - 3) - 5 x$

Group like terms:

$- x - 1 = (5 x - 5 x) - 3$

Simplify the arithmetic:

$- x - 1 = - 3$

Add to both sides:

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

Simplify the arithmetic:

$- x = - 3 + 1$

Simplify the arithmetic:

$- x = - 2$

Multiply both sides by :

$- x \cdot - 1 = - 2 \cdot - 1$

Remove the one(s):

$x = - 2 \cdot - 1$

Simplify the arithmetic:

$x = 2$

10 additional steps

$(4 x - 1) = - (5 x - 3)$

Expand the parentheses:

$(4 x - 1) = - 5 x + 3$

Add to both sides:

$(4 x - 1) + 5 x = (- 5 x + 3) + 5 x$

Group like terms:

$(4 x + 5 x) - 1 = (- 5 x + 3) + 5 x$

Simplify the arithmetic:

$9 x - 1 = (- 5 x + 3) + 5 x$

Group like terms:

$9 x - 1 = (- 5 x + 5 x) + 3$

Simplify the arithmetic:

$9 x - 1 = 3$

Add to both sides:

$(9 x - 1) + 1 = 3 + 1$

Simplify the arithmetic:

$9 x = 3 + 1$

Simplify the arithmetic:

$9 x = 4$

Divide both sides by :

$\frac{(9 x)}{9} = \frac{4}{9}$

Simplify the fraction:

$x = \frac{4}{9}$

3. List the solutions

$x = 2, \frac{4}{9}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | 4 x - 1 |$
$y = | 5 x - 3 |$
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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