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Solution - Absolute value equations

Exact form:

x = \frac{6}{7}, 4

x=\frac{6}{7} , 4

Decimal form:

x = 0.857, 4

x=0.857 , 4

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

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

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:

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

2. Solve the two equations for x

9 additional steps

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

Add to both sides:

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

Group like terms:

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

Simplify the arithmetic:

$7 x - 1 = (- 4 x + 5) + 4 x$

Group like terms:

$7 x - 1 = (- 4 x + 4 x) + 5$

Simplify the arithmetic:

$7 x - 1 = 5$

Add to both sides:

$(7 x - 1) + 1 = 5 + 1$

Simplify the arithmetic:

$7 x = 5 + 1$

Simplify the arithmetic:

$7 x = 6$

Divide both sides by :

$\frac{(7 x)}{7} = \frac{6}{7}$

Simplify the fraction:

$x = \frac{6}{7}$

11 additional steps

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

Expand the parentheses:

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

Subtract from both sides:

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

Group like terms:

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

Simplify the arithmetic:

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

Group like terms:

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

Simplify the arithmetic:

$- x - 1 = - 5$

Add to both sides:

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

Simplify the arithmetic:

$- x = - 5 + 1$

Simplify the arithmetic:

$- x = - 4$

Multiply both sides by :

$- x \cdot - 1 = - 4 \cdot - 1$

Remove the one(s):

$x = - 4 \cdot - 1$

Simplify the arithmetic:

$x = 4$

3. List the solutions

$x = \frac{6}{7}, 4$
(2 solution(s))

4. Graph

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