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Solution - Absolute value equations

Exact form:

x = \frac{7}{4}, \frac{13}{8}

x=\frac{7}{4} , \frac{13}{8}

Mixed number form:

x = 1 \frac{3}{4}, 1 \frac{5}{8}

x=1\frac{3}{4} , 1\frac{5}{8}

Decimal form:

x = 1.75, 1.625

x=1.75 , 1.625

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
$| 2 x - 3 | = 2 | 3 x - 5 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| 2 x - 3 \| = 2 \| 3 x - 5 \|$
$x = + y$	$(2 x - 3) = 2 (3 x - 5)$
$x = - y$	$(2 x - 3) = 2 (- (3 x - 5))$
$+ x = y$	$(2 x - 3) = 2 (3 x - 5)$
$- x = y$	$- (2 x - 3) = 2 (3 x - 5)$

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:

$\| x \| = \| y \|$	$\| 2 x - 3 \| = 2 \| 3 x - 5 \|$
$x = + y$ , $+ x = y$	$(2 x - 3) = 2 (3 x - 5)$
$x = - y$ , $- x = y$	$(2 x - 3) = 2 (- (3 x - 5))$

2. Solve the two equations for x

14 additional steps

$(2 x - 3) = 2 \cdot (3 x - 5)$

Expand the parentheses:

$(2 x - 3) = 2 \cdot 3 x + 2 \cdot - 5$

Multiply the coefficients:

$(2 x - 3) = 6 x + 2 \cdot - 5$

Simplify the arithmetic:

$(2 x - 3) = 6 x - 10$

Subtract from both sides:

$(2 x - 3) - 6 x = (6 x - 10) - 6 x$

Group like terms:

$(2 x - 6 x) - 3 = (6 x - 10) - 6 x$

Simplify the arithmetic:

$- 4 x - 3 = (6 x - 10) - 6 x$

Group like terms:

$- 4 x - 3 = (6 x - 6 x) - 10$

Simplify the arithmetic:

$- 4 x - 3 = - 10$

Add to both sides:

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

Simplify the arithmetic:

$- 4 x = - 10 + 3$

Simplify the arithmetic:

$- 4 x = - 7$

Divide both sides by :

$\frac{(- 4 x)}{- 4} = \frac{- 7}{- 4}$

Cancel out the negatives:

$\frac{4 x}{4} = \frac{- 7}{- 4}$

Simplify the fraction:

$x = \frac{- 7}{- 4}$

Cancel out the negatives:

$x = \frac{7}{4}$

13 additional steps

$(2 x - 3) = 2 \cdot (- (3 x - 5))$

Expand the parentheses:

$(2 x - 3) = 2 \cdot (- 3 x + 5)$

Expand the parentheses:

$(2 x - 3) = 2 \cdot - 3 x + 2 \cdot 5$

Multiply the coefficients:

$(2 x - 3) = - 6 x + 2 \cdot 5$

Simplify the arithmetic:

$(2 x - 3) = - 6 x + 10$

Add to both sides:

$(2 x - 3) + 6 x = (- 6 x + 10) + 6 x$

Group like terms:

$(2 x + 6 x) - 3 = (- 6 x + 10) + 6 x$

Simplify the arithmetic:

$8 x - 3 = (- 6 x + 10) + 6 x$

Group like terms:

$8 x - 3 = (- 6 x + 6 x) + 10$

Simplify the arithmetic:

$8 x - 3 = 10$

Add to both sides:

$(8 x - 3) + 3 = 10 + 3$

Simplify the arithmetic:

$8 x = 10 + 3$

Simplify the arithmetic:

$8 x = 13$

Divide both sides by :

$\frac{(8 x)}{8} = \frac{13}{8}$

Simplify the fraction:

$x = \frac{13}{8}$

3. List the solutions

$x = \frac{7}{4}, \frac{13}{8}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | 2 x - 3 |$
$y = 2 | 3 x - 5 |$
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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