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Solution - Absolute value equations

Exact form:

u = - 13, - \frac{7}{3}

u=-13 , -\frac{7}{3}

Mixed number form:

u = - 13, - 2 \frac{1}{3}

u=-13 , -2\frac{1}{3}

Decimal form:

u = - 13, - 2.333

u=-13 , -2.333

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
$| u - 3 | = | 2 u + 10 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| u - 3 \| = \| 2 u + 10 \|$
$x = + y$	$(u - 3) = (2 u + 10)$
$x = - y$	$(u - 3) = - (2 u + 10)$
$+ x = y$	$(u - 3) = (2 u + 10)$
$- x = y$	$- (u - 3) = (2 u + 10)$

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:

$\| x \| = \| y \|$	$\| u - 3 \| = \| 2 u + 10 \|$
$x = + y$ , $+ x = y$	$(u - 3) = (2 u + 10)$
$x = - y$ , $- x = y$	$(u - 3) = - (2 u + 10)$

2. Solve the two equations for u

10 additional steps

$(u - 3) = (2 u + 10)$

Subtract from both sides:

$(u - 3) - 2 u = (2 u + 10) - 2 u$

Group like terms:

$(u - 2 u) - 3 = (2 u + 10) - 2 u$

Simplify the arithmetic:

$- u - 3 = (2 u + 10) - 2 u$

Group like terms:

$- u - 3 = (2 u - 2 u) + 10$

Simplify the arithmetic:

$- u - 3 = 10$

Add to both sides:

$(- u - 3) + 3 = 10 + 3$

Simplify the arithmetic:

$- u = 10 + 3$

Simplify the arithmetic:

$- u = 13$

Multiply both sides by :

$- u \cdot - 1 = 13 \cdot - 1$

Remove the one(s):

$u = 13 \cdot - 1$

Simplify the arithmetic:

$u = - 13$

10 additional steps

$(u - 3) = - (2 u + 10)$

Expand the parentheses:

$(u - 3) = - 2 u - 10$

Add to both sides:

$(u - 3) + 2 u = (- 2 u - 10) + 2 u$

Group like terms:

$(u + 2 u) - 3 = (- 2 u - 10) + 2 u$

Simplify the arithmetic:

$3 u - 3 = (- 2 u - 10) + 2 u$

Group like terms:

$3 u - 3 = (- 2 u + 2 u) - 10$

Simplify the arithmetic:

$3 u - 3 = - 10$

Add to both sides:

$(3 u - 3) + 3 = - 10 + 3$

Simplify the arithmetic:

$3 u = - 10 + 3$

Simplify the arithmetic:

$3 u = - 7$

Divide both sides by :

$\frac{(3 u)}{3} = \frac{- 7}{3}$

Simplify the fraction:

$u = \frac{- 7}{3}$

3. List the solutions

$u = - 13, - \frac{7}{3}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | u - 3 |$
$y = | 2 u + 10 |$
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 u

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 u

3. List the solutions

4. Graph

Why learn this

Terms and topics

Related links

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