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Solution - Absolute value equations

Exact form:

x = \frac{8}{13}, \frac{12}{17}

x=\frac{8}{13} , \frac{12}{17}

Decimal form:

x = 0.615, 0.706

x=0.615 , 0.706

See steps

Step-by-step explanation

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

$5 | 3 x - 2 | - 2 | x - 1 | = 0$

Add $2 | x - 1 |$ to both sides of the equation:

$5 | 3 x - 2 | - 2 | x - 1 | + 2 | x - 1 | = 2 | x - 1 |$

Simplify the arithmetic

$5 | 3 x - 2 | = 2 | x - 1 |$

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
$5 | 3 x - 2 | = 2 | x - 1 |$
without the absolute value bars:

$\| x \| = \| y \|$	$5 \| 3 x - 2 \| = 2 \| x - 1 \|$
$x = + y$	$5 (3 x - 2) = 2 (x - 1)$
$x = - y$	$5 (3 x - 2) = 2 (- (x - 1))$
$+ x = y$	$5 (3 x - 2) = 2 (x - 1)$
$- x = y$	$5 (- (3 x - 2)) = 2 (x - 1)$

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:

$\| x \| = \| y \|$	$5 \| 3 x - 2 \| = 2 \| x - 1 \|$
$x = + y$ , $+ x = y$	$5 (3 x - 2) = 2 (x - 1)$
$x = - y$ , $- x = y$	$5 (3 x - 2) = 2 (- (x - 1))$

3. Solve the two equations for x

14 additional steps

$5 \cdot (3 x - 2) = 2 \cdot (x - 1)$

Expand the parentheses:

$5 \cdot 3 x + 5 \cdot - 2 = 2 \cdot (x - 1)$

Multiply the coefficients:

$15 x + 5 \cdot - 2 = 2 \cdot (x - 1)$

Simplify the arithmetic:

$15 x - 10 = 2 \cdot (x - 1)$

Expand the parentheses:

$15 x - 10 = 2 x + 2 \cdot - 1$

Simplify the arithmetic:

$15 x - 10 = 2 x - 2$

Subtract from both sides:

$(15 x - 10) - 2 x = (2 x - 2) - 2 x$

Group like terms:

$(15 x - 2 x) - 10 = (2 x - 2) - 2 x$

Simplify the arithmetic:

$13 x - 10 = (2 x - 2) - 2 x$

Group like terms:

$13 x - 10 = (2 x - 2 x) - 2$

Simplify the arithmetic:

$13 x - 10 = - 2$

Add to both sides:

$(13 x - 10) + 10 = - 2 + 10$

Simplify the arithmetic:

$13 x = - 2 + 10$

Simplify the arithmetic:

$13 x = 8$

Divide both sides by :

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

Simplify the fraction:

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

17 additional steps

$5 \cdot (3 x - 2) = 2 \cdot (- (x - 1))$

Expand the parentheses:

$5 \cdot 3 x + 5 \cdot - 2 = 2 \cdot (- (x - 1))$

Multiply the coefficients:

$15 x + 5 \cdot - 2 = 2 \cdot (- (x - 1))$

Simplify the arithmetic:

$15 x - 10 = 2 \cdot (- (x - 1))$

Expand the parentheses:

$15 x - 10 = 2 \cdot (- x + 1)$

$15 x - 10 = 2 \cdot - x + 2 \cdot 1$

Group like terms:

$15 x - 10 = (2 \cdot - 1) x + 2 \cdot 1$

Multiply the coefficients:

$15 x - 10 = - 2 x + 2 \cdot 1$

Simplify the arithmetic:

$15 x - 10 = - 2 x + 2$

Add to both sides:

$(15 x - 10) + 2 x = (- 2 x + 2) + 2 x$

Group like terms:

$(15 x + 2 x) - 10 = (- 2 x + 2) + 2 x$

Simplify the arithmetic:

$17 x - 10 = (- 2 x + 2) + 2 x$

Group like terms:

$17 x - 10 = (- 2 x + 2 x) + 2$

Simplify the arithmetic:

$17 x - 10 = 2$

Add to both sides:

$(17 x - 10) + 10 = 2 + 10$

Simplify the arithmetic:

$17 x = 2 + 10$

Simplify the arithmetic:

$17 x = 12$

Divide both sides by :

$\frac{(17 x)}{17} = \frac{12}{17}$

Simplify the fraction:

$x = \frac{12}{17}$

4. List the solutions

$x = \frac{8}{13}, \frac{12}{17}$
(2 solution(s))

5. Graph

Each line represents the function of one side of the equation:
$y = 5 | 3 x - 2 |$
$y = 2 | x - 1 |$
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