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Solution - Absolute value equations

Exact form:

x = 4, - \frac{2}{5}

x=4 , -\frac{2}{5}

Decimal form:

x = 4, - 0.4

x=4 , -0.4

See steps

Step-by-step explanation

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

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

Add $- | 3 x - 1 |$ to both sides of the equation:

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

Simplify the arithmetic

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

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

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

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:

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

3. Solve the two equations for x

8 additional steps

$(- 2 x - 3) = - (3 x - 1)$

Expand the parentheses:

$(- 2 x - 3) = - 3 x + 1$

Add to both sides:

$(- 2 x - 3) + 3 x = (- 3 x + 1) + 3 x$

Group like terms:

$(- 2 x + 3 x) - 3 = (- 3 x + 1) + 3 x$

Simplify the arithmetic:

$x - 3 = (- 3 x + 1) + 3 x$

Group like terms:

$x - 3 = (- 3 x + 3 x) + 1$

Simplify the arithmetic:

$x - 3 = 1$

Add to both sides:

$(x - 3) + 3 = 1 + 3$

Simplify the arithmetic:

$x = 1 + 3$

Simplify the arithmetic:

$x = 4$

12 additional steps

$(- 2 x - 3) = - (- (3 x - 1))$

NT_MSLUS_MAINSTEP_RESOLVE_DOUBLE_MINUS:

$(- 2 x - 3) = 3 x - 1$

Subtract from both sides:

$(- 2 x - 3) - 3 x = (3 x - 1) - 3 x$

Group like terms:

$(- 2 x - 3 x) - 3 = (3 x - 1) - 3 x$

Simplify the arithmetic:

$- 5 x - 3 = (3 x - 1) - 3 x$

Group like terms:

$- 5 x - 3 = (3 x - 3 x) - 1$

Simplify the arithmetic:

$- 5 x - 3 = - 1$

Add to both sides:

$(- 5 x - 3) + 3 = - 1 + 3$

Simplify the arithmetic:

$- 5 x = - 1 + 3$

Simplify the arithmetic:

$- 5 x = 2$

Divide both sides by :

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

Cancel out the negatives:

$\frac{5 x}{5} = \frac{2}{- 5}$

Simplify the fraction:

$x = \frac{2}{- 5}$

Move the negative sign from the denominator to the numerator:

$x = \frac{- 2}{5}$

4. List the solutions

$x = 4, - \frac{2}{5}$
(2 solution(s))

5. Graph

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