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Solution - Absolute value equations

Exact form:

x = 2, 2

x=2 , 2

See steps

Step-by-step explanation

1. Rewrite the equation with one absolute value terms on each side

$| 5 x - 10 | + | 2 x - 4 | = 0$

Add $- | 2 x - 4 |$ to both sides of the equation:

$| 5 x - 10 | + | 2 x - 4 | - | 2 x - 4 | = - | 2 x - 4 |$

Simplify the arithmetic

$| 5 x - 10 | = - | 2 x - 4 |$

2. 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
$| 5 x - 10 | = - | 2 x - 4 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| 5 x - 10 \| = - \| 2 x - 4 \|$
$x = + y$	$(5 x - 10) = - (2 x - 4)$
$x = - y$	$(5 x - 10) = - - (2 x - 4)$
$+ x = y$	$(5 x - 10) = - (2 x - 4)$
$- x = y$	$- (5 x - 10) = - (2 x - 4)$

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:

$\| x \| = \| y \|$	$\| 5 x - 10 \| = - \| 2 x - 4 \|$
$x = + y$ , $+ x = y$	$(5 x - 10) = - (2 x - 4)$
$x = - y$ , $- x = y$	$(5 x - 10) = - - (2 x - 4)$

3. Solve the two equations for x

12 additional steps

$(5 x - 10) = - (2 x - 4)$

Expand the parentheses:

$(5 x - 10) = - 2 x + 4$

Add to both sides:

$(5 x - 10) + 2 x = (- 2 x + 4) + 2 x$

Group like terms:

$(5 x + 2 x) - 10 = (- 2 x + 4) + 2 x$

Simplify the arithmetic:

$7 x - 10 = (- 2 x + 4) + 2 x$

Group like terms:

$7 x - 10 = (- 2 x + 2 x) + 4$

Simplify the arithmetic:

$7 x - 10 = 4$

Add to both sides:

$(7 x - 10) + 10 = 4 + 10$

Simplify the arithmetic:

$7 x = 4 + 10$

Simplify the arithmetic:

$7 x = 14$

Divide both sides by :

$\frac{(7 x)}{7} = \frac{14}{7}$

Simplify the fraction:

$x = \frac{14}{7}$

Find the greatest common factor of the numerator and denominator:

$x = \frac{(2 \cdot 7)}{(1 \cdot 7)}$

Factor out and cancel the greatest common factor:

$x = 2$

12 additional steps

$(5 x - 10) = - (- (2 x - 4))$

NT_MSLUS_MAINSTEP_RESOLVE_DOUBLE_MINUS:

$(5 x - 10) = 2 x - 4$

Subtract from both sides:

$(5 x - 10) - 2 x = (2 x - 4) - 2 x$

Group like terms:

$(5 x - 2 x) - 10 = (2 x - 4) - 2 x$

Simplify the arithmetic:

$3 x - 10 = (2 x - 4) - 2 x$

Group like terms:

$3 x - 10 = (2 x - 2 x) - 4$

Simplify the arithmetic:

$3 x - 10 = - 4$

Add to both sides:

$(3 x - 10) + 10 = - 4 + 10$

Simplify the arithmetic:

$3 x = - 4 + 10$

Simplify the arithmetic:

$3 x = 6$

Divide both sides by :

$\frac{(3 x)}{3} = \frac{6}{3}$

Simplify the fraction:

$x = \frac{6}{3}$

Find the greatest common factor of the numerator and denominator:

$x = \frac{(2 \cdot 3)}{(1 \cdot 3)}$

Factor out and cancel the greatest common factor:

$x = 2$

4. List the solutions

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

5. Graph

Each line represents the function of one side of the equation:
$y = | 5 x - 10 |$
$y = - | 2 x - 4 |$
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 with one absolute value terms on each side

2. Rewrite the equation without absolute value bars

3. Solve the two equations for x

4. List the solutions

5. Graph

Why learn this

Terms and topics

Related links

Solution - Absolute value equations

Step-by-step explanation

1. Rewrite the equation with one absolute value terms on each side

2. Rewrite the equation without absolute value bars

3. Solve the two equations for x

4. List the solutions

5. Graph

Why learn this

Terms and topics

Related links

Latest Related Drills Solved