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Solution - Absolute value equations

Exact form:

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

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

Decimal form:

x = 0.667, 0.667

x=0.667 , 0.667

See steps

Step-by-step explanation

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

$| 3 x - 2 | - 3 | 3 x - 2 | = 0$

Add $3 | 3 x - 2 |$ to both sides of the equation:

$| 3 x - 2 | - 3 | 3 x - 2 | + 3 | 3 x - 2 | = 3 | 3 x - 2 |$

Simplify the arithmetic

$| 3 x - 2 | = 3 | 3 x - 2 |$

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

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

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:

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

3. Solve the two equations for x

16 additional steps

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

Expand the parentheses:

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

Multiply the coefficients:

$(3 x - 2) = 9 x + 3 \cdot - 2$

Simplify the arithmetic:

$(3 x - 2) = 9 x - 6$

Subtract from both sides:

$(3 x - 2) - 9 x = (9 x - 6) - 9 x$

Group like terms:

$(3 x - 9 x) - 2 = (9 x - 6) - 9 x$

Simplify the arithmetic:

$- 6 x - 2 = (9 x - 6) - 9 x$

Group like terms:

$- 6 x - 2 = (9 x - 9 x) - 6$

Simplify the arithmetic:

$- 6 x - 2 = - 6$

Add to both sides:

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

Simplify the arithmetic:

$- 6 x = - 6 + 2$

Simplify the arithmetic:

$- 6 x = - 4$

Divide both sides by :

$\frac{(- 6 x)}{- 6} = \frac{- 4}{- 6}$

Cancel out the negatives:

$\frac{6 x}{6} = \frac{- 4}{- 6}$

Simplify the fraction:

$x = \frac{- 4}{- 6}$

Cancel out the negatives:

$x = \frac{4}{6}$

Find the greatest common factor of the numerator and denominator:

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

Factor out and cancel the greatest common factor:

$x = \frac{2}{3}$

15 additional steps

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

Expand the parentheses:

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

Expand the parentheses:

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

Multiply the coefficients:

$(3 x - 2) = - 9 x + 3 \cdot 2$

Simplify the arithmetic:

$(3 x - 2) = - 9 x + 6$

Add to both sides:

$(3 x - 2) + 9 x = (- 9 x + 6) + 9 x$

Group like terms:

$(3 x + 9 x) - 2 = (- 9 x + 6) + 9 x$

Simplify the arithmetic:

$12 x - 2 = (- 9 x + 6) + 9 x$

Group like terms:

$12 x - 2 = (- 9 x + 9 x) + 6$

Simplify the arithmetic:

$12 x - 2 = 6$

Add to both sides:

$(12 x - 2) + 2 = 6 + 2$

Simplify the arithmetic:

$12 x = 6 + 2$

Simplify the arithmetic:

$12 x = 8$

Divide both sides by :

$\frac{(12 x)}{12} = \frac{8}{12}$

Simplify the fraction:

$x = \frac{8}{12}$

Find the greatest common factor of the numerator and denominator:

$x = \frac{(2 \cdot 4)}{(3 \cdot 4)}$

Factor out and cancel the greatest common factor:

$x = \frac{2}{3}$

4. List the solutions

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

5. Graph

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