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Solution - Absolute value equations

Exact form:

x = - 2, - \frac{16}{5}

x=-2 , -\frac{16}{5}

Mixed number form:

x = - 2, - 3 \frac{1}{5}

x=-2 , -3\frac{1}{5}

Decimal form:

x = - 2, - 3.2

x=-2 , -3.2

See steps

Step-by-step explanation

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

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

Add $- 3 | 2 x + 5 |$ to both sides of the equation:

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

Simplify the arithmetic

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

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

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

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:

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

3. Solve the two equations for x

14 additional steps

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

Expand the parentheses:

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

Multiply the coefficients:

$(x - 1) = - 6 x - 3 \cdot 5$

Simplify the arithmetic:

$(x - 1) = - 6 x - 15$

Add to both sides:

$(x - 1) + 6 x = (- 6 x - 15) + 6 x$

Group like terms:

$(x + 6 x) - 1 = (- 6 x - 15) + 6 x$

Simplify the arithmetic:

$7 x - 1 = (- 6 x - 15) + 6 x$

Group like terms:

$7 x - 1 = (- 6 x + 6 x) - 15$

Simplify the arithmetic:

$7 x - 1 = - 15$

Add to both sides:

$(7 x - 1) + 1 = - 15 + 1$

Simplify the arithmetic:

$7 x = - 15 + 1$

Simplify the arithmetic:

$7 x = - 14$

Divide both sides by :

$\frac{(7 x)}{7} = \frac{- 14}{7}$

Simplify the fraction:

$x = \frac{- 14}{7}$

Find the greatest common factor of the numerator and denominator:

$x = \frac{(- 2 \cdot 7)}{(1 \cdot 7)}$

Factor out and cancel the greatest common factor:

$x = - 2$

15 additional steps

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

Expand the parentheses:

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

Expand the parentheses:

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

Multiply the coefficients:

$(x - 1) = 6 x - 3 \cdot - 5$

Simplify the arithmetic:

$(x - 1) = 6 x + 15$

Subtract from both sides:

$(x - 1) - 6 x = (6 x + 15) - 6 x$

Group like terms:

$(x - 6 x) - 1 = (6 x + 15) - 6 x$

Simplify the arithmetic:

$- 5 x - 1 = (6 x + 15) - 6 x$

Group like terms:

$- 5 x - 1 = (6 x - 6 x) + 15$

Simplify the arithmetic:

$- 5 x - 1 = 15$

Add to both sides:

$(- 5 x - 1) + 1 = 15 + 1$

Simplify the arithmetic:

$- 5 x = 15 + 1$

Simplify the arithmetic:

$- 5 x = 16$

Divide both sides by :

$\frac{(- 5 x)}{- 5} = \frac{16}{- 5}$

Cancel out the negatives:

$\frac{5 x}{5} = \frac{16}{- 5}$

Simplify the fraction:

$x = \frac{16}{- 5}$

Move the negative sign from the denominator to the numerator:

$x = \frac{- 16}{5}$

4. List the solutions

$x = - 2, - \frac{16}{5}$
(2 solution(s))

5. Graph

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