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Solution - Absolute value equations

Exact form:

u = \frac{4}{3}, - 6

u=\frac{4}{3} , -6

Mixed number form:

u = 1 \frac{1}{3}, - 6

u=1\frac{1}{3} , -6

Decimal form:

u = 1.333, - 6

u=1.333 , -6

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

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

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:

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

2. Solve the two equations for u

9 additional steps

$(2 u + 1) = (- u + 5)$

Add to both sides:

$(2 u + 1) + u = (- u + 5) + u$

Group like terms:

$(2 u + u) + 1 = (- u + 5) + u$

Simplify the arithmetic:

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

Group like terms:

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

Simplify the arithmetic:

$3 u + 1 = 5$

Subtract from both sides:

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

Simplify the arithmetic:

$3 u = 5 - 1$

Simplify the arithmetic:

$3 u = 4$

Divide both sides by :

$\frac{(3 u)}{3} = \frac{4}{3}$

Simplify the fraction:

$u = \frac{4}{3}$

8 additional steps

$(2 u + 1) = - (- u + 5)$

Expand the parentheses:

$(2 u + 1) = u - 5$

Subtract from both sides:

$(2 u + 1) - u = (u - 5) - u$

Group like terms:

$(2 u - u) + 1 = (u - 5) - u$

Simplify the arithmetic:

$u + 1 = (u - 5) - u$

Group like terms:

$u + 1 = (u - u) - 5$

Simplify the arithmetic:

$u + 1 = - 5$

Subtract from both sides:

$(u + 1) - 1 = - 5 - 1$

Simplify the arithmetic:

$u = - 5 - 1$

Simplify the arithmetic:

$u = - 6$

3. List the solutions

$u = \frac{4}{3}, - 6$
(2 solution(s))

4. Graph

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