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Solution - Absolute value equations

Exact form:

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

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

Mixed number form:

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

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

Decimal form:

x = 1.667, 1.667

x=1.667 , 1.667

See steps

Step-by-step explanation

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

$| - 3 x + 5 | + | 2 x - \frac{10}{3} | = 0$

Add $- | 2 x - \frac{10}{3} |$ to both sides of the equation:

$| - 3 x + 5 | + | 2 x - \frac{10}{3} | - | 2 x - \frac{10}{3} | = - | 2 x - \frac{10}{3} |$

Simplify the arithmetic

$| - 3 x + 5 | = - | 2 x - \frac{10}{3} |$

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
$| - 3 x + 5 | = - | 2 x - \frac{10}{3} |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| - 3 x + 5 \| = - \| 2 x - \frac{10}{3} \|$
$x = + y$	$(- 3 x + 5) = - (2 x - \frac{10}{3})$
$x = - y$	$(- 3 x + 5) = - - (2 x - \frac{10}{3})$
$+ x = y$	$(- 3 x + 5) = - (2 x - \frac{10}{3})$
$- x = y$	$- (- 3 x + 5) = - (2 x - \frac{10}{3})$

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:

$\| x \| = \| y \|$	$\| - 3 x + 5 \| = - \| 2 x - \frac{10}{3} \|$
$x = + y$ , $+ x = y$	$(- 3 x + 5) = - (2 x - \frac{10}{3})$
$x = - y$ , $- x = y$	$(- 3 x + 5) = - - (2 x - \frac{10}{3})$

3. Solve the two equations for x

13 additional steps

$(- 3 x + 5) = - (2 x + \frac{- 10}{3})$

Expand the parentheses:

$(- 3 x + 5) = - 2 x + \frac{10}{3}$

Add to both sides:

$(- 3 x + 5) + 2 x = (- 2 x + \frac{10}{3}) + 2 x$

Group like terms:

$(- 3 x + 2 x) + 5 = (- 2 x + \frac{10}{3}) + 2 x$

Simplify the arithmetic:

$- x + 5 = (- 2 x + \frac{10}{3}) + 2 x$

Group like terms:

$- x + 5 = (- 2 x + 2 x) + \frac{10}{3}$

Simplify the arithmetic:

$- x + 5 = \frac{10}{3}$

Subtract from both sides:

$(- x + 5) - 5 = (\frac{10}{3}) - 5$

Simplify the arithmetic:

$- x = (\frac{10}{3}) - 5$

Convert the integer into a fraction:

$- x = \frac{10}{3} + \frac{- 15}{3}$

Combine the fractions:

$- x = \frac{(10 - 15)}{3}$

Combine the numerators:

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

Multiply both sides by :

$- x \cdot - 1 = (\frac{- 5}{3}) \cdot - 1$

Remove the one(s):

$x = (\frac{- 5}{3}) \cdot - 1$

Remove the one(s):

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

15 additional steps

$(- 3 x + 5) = - (- (2 x + \frac{- 10}{3}))$

NT_MSLUS_MAINSTEP_RESOLVE_DOUBLE_MINUS:

$(- 3 x + 5) = 2 x + \frac{- 10}{3}$

Subtract from both sides:

$(- 3 x + 5) - 2 x = (2 x + \frac{- 10}{3}) - 2 x$

Group like terms:

$(- 3 x - 2 x) + 5 = (2 x + \frac{- 10}{3}) - 2 x$

Simplify the arithmetic:

$- 5 x + 5 = (2 x + \frac{- 10}{3}) - 2 x$

Group like terms:

$- 5 x + 5 = (2 x - 2 x) + \frac{- 10}{3}$

Simplify the arithmetic:

$- 5 x + 5 = \frac{- 10}{3}$

Subtract from both sides:

$(- 5 x + 5) - 5 = (\frac{- 10}{3}) - 5$

Simplify the arithmetic:

$- 5 x = (\frac{- 10}{3}) - 5$

Convert the integer into a fraction:

$- 5 x = \frac{- 10}{3} + \frac{- 15}{3}$

Combine the fractions:

$- 5 x = \frac{(- 10 - 15)}{3}$

Combine the numerators:

$- 5 x = \frac{- 25}{3}$

Divide both sides by :

$\frac{(- 5 x)}{- 5} = \frac{(\frac{- 25}{3})}{- 5}$

Cancel out the negatives:

$\frac{5 x}{5} = \frac{(\frac{- 25}{3})}{- 5}$

Simplify the fraction:

$x = \frac{(\frac{- 25}{3})}{- 5}$

Simplify the arithmetic:

$x = \frac{- 25}{(3 \cdot - 5)}$

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

4. List the solutions

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

5. Graph

Each line represents the function of one side of the equation:
$y = | - 3 x + 5 |$
$y = - | 2 x - \frac{10}{3} |$
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 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