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Solution - Absolute value equations

Exact form:

u = 10, - \frac{2}{3}

u=10 , -\frac{2}{3}

Decimal form:

u = 10, - 0.667

u=10 , -0.667

See steps

Step-by-step explanation

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

$| 2 u - 4 | - | u + 6 | = 0$

Add $| u + 6 |$ to both sides of the equation:

$| 2 u - 4 | - | u + 6 | + | u + 6 | = | u + 6 |$

Simplify the arithmetic

$| 2 u - 4 | = | u + 6 |$

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
$| 2 u - 4 | = | u + 6 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| 2 u - 4 \| = \| u + 6 \|$
$x = + y$	$(2 u - 4) = (u + 6)$
$x = - y$	$(2 u - 4) = (- (u + 6))$
$+ x = y$	$(2 u - 4) = (u + 6)$
$- x = y$	$- (2 u - 4) = (u + 6)$

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:

$\| x \| = \| y \|$	$\| 2 u - 4 \| = \| u + 6 \|$
$x = + y$ , $+ x = y$	$(2 u - 4) = (u + 6)$
$x = - y$ , $- x = y$	$(2 u - 4) = (- (u + 6))$

3. Solve the two equations for u

7 additional steps

$(2 u - 4) = (u + 6)$

Subtract from both sides:

$(2 u - 4) - u = (u + 6) - u$

Group like terms:

$(2 u - u) - 4 = (u + 6) - u$

Simplify the arithmetic:

$u - 4 = (u + 6) - u$

Group like terms:

$u - 4 = (u - u) + 6$

Simplify the arithmetic:

$u - 4 = 6$

Add to both sides:

$(u - 4) + 4 = 6 + 4$

Simplify the arithmetic:

$u = 6 + 4$

Simplify the arithmetic:

$u = 10$

10 additional steps

$(2 u - 4) = - (u + 6)$

Expand the parentheses:

$(2 u - 4) = - u - 6$

Add to both sides:

$(2 u - 4) + u = (- u - 6) + u$

Group like terms:

$(2 u + u) - 4 = (- u - 6) + u$

Simplify the arithmetic:

$3 u - 4 = (- u - 6) + u$

Group like terms:

$3 u - 4 = (- u + u) - 6$

Simplify the arithmetic:

$3 u - 4 = - 6$

Add to both sides:

$(3 u - 4) + 4 = - 6 + 4$

Simplify the arithmetic:

$3 u = - 6 + 4$

Simplify the arithmetic:

$3 u = - 2$

Divide both sides by :

$\frac{(3 u)}{3} = \frac{- 2}{3}$

Simplify the fraction:

$u = \frac{- 2}{3}$

4. List the solutions

$u = 10, - \frac{2}{3}$
(2 solution(s))

5. Graph

Each line represents the function of one side of the equation:
$y = | 2 u - 4 |$
$y = | u + 6 |$
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 u

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 u

4. List the solutions

5. Graph

Why learn this

Terms and topics

Related links

Latest Related Drills Solved