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Solution - Absolute value equations

Exact form:

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

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

Mixed number form:

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

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

Decimal form:

x = 0.5, 1.667

x=0.5 , 1.667

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
$| 5 x - 6 | = | x - 4 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| 5 x - 6 \| = \| x - 4 \|$
$x = + y$	$(5 x - 6) = (x - 4)$
$x = - y$	$(5 x - 6) = - (x - 4)$
$+ x = y$	$(5 x - 6) = (x - 4)$
$- x = y$	$- (5 x - 6) = (x - 4)$

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:

$\| x \| = \| y \|$	$\| 5 x - 6 \| = \| x - 4 \|$
$x = + y$ , $+ x = y$	$(5 x - 6) = (x - 4)$
$x = - y$ , $- x = y$	$(5 x - 6) = - (x - 4)$

2. Solve the two equations for x

11 additional steps

$(5 x - 6) = (x - 4)$

Subtract from both sides:

$(5 x - 6) - x = (x - 4) - x$

Group like terms:

$(5 x - x) - 6 = (x - 4) - x$

Simplify the arithmetic:

$4 x - 6 = (x - 4) - x$

Group like terms:

$4 x - 6 = (x - x) - 4$

Simplify the arithmetic:

$4 x - 6 = - 4$

Add to both sides:

$(4 x - 6) + 6 = - 4 + 6$

Simplify the arithmetic:

$4 x = - 4 + 6$

Simplify the arithmetic:

$4 x = 2$

Divide both sides by :

$\frac{(4 x)}{4} = \frac{2}{4}$

Simplify the fraction:

$x = \frac{2}{4}$

Find the greatest common factor of the numerator and denominator:

$x = \frac{(1 \cdot 2)}{(2 \cdot 2)}$

Factor out and cancel the greatest common factor:

$x = \frac{1}{2}$

12 additional steps

$(5 x - 6) = - (x - 4)$

Expand the parentheses:

$(5 x - 6) = - x + 4$

Add to both sides:

$(5 x - 6) + x = (- x + 4) + x$

Group like terms:

$(5 x + x) - 6 = (- x + 4) + x$

Simplify the arithmetic:

$6 x - 6 = (- x + 4) + x$

Group like terms:

$6 x - 6 = (- x + x) + 4$

Simplify the arithmetic:

$6 x - 6 = 4$

Add to both sides:

$(6 x - 6) + 6 = 4 + 6$

Simplify the arithmetic:

$6 x = 4 + 6$

Simplify the arithmetic:

$6 x = 10$

Divide both sides by :

$\frac{(6 x)}{6} = \frac{10}{6}$

Simplify the fraction:

$x = \frac{10}{6}$

Find the greatest common factor of the numerator and denominator:

$x = \frac{(5 \cdot 2)}{(3 \cdot 2)}$

Factor out and cancel the greatest common factor:

$x = \frac{5}{3}$

3. List the solutions

$x = \frac{1}{2}, \frac{5}{3}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | 5 x - 6 |$
$y = | x - 4 |$
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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