Enter an equation or problem

Camera input is not recognized!

Solution - Absolute value equations

Exact form:

u = \frac{3}{2}

u=\frac{3}{2}

Mixed number form:

u = 1 \frac{1}{2}

u=1\frac{1}{2}

Decimal form:

u = 1.5

u=1.5

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

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

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:

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

2. Solve the two equations for u

4 additional steps

$(2 u - 6) = 2 u$

Subtract from both sides:

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

Group like terms:

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

Simplify the arithmetic:

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

Simplify the arithmetic:

$- 6 = 0$

The statement is false:

$- 6 = 0$

The equation is false so it has no solution.

9 additional steps

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

Add to both sides:

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

Simplify the arithmetic:

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

Add to both sides:

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

Simplify the arithmetic:

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

Group like terms:

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

Simplify the arithmetic:

$4 u = 6$

Divide both sides by :

$\frac{(4 u)}{4} = \frac{6}{4}$

Simplify the fraction:

$u = \frac{6}{4}$

Find the greatest common factor of the numerator and denominator:

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

Factor out and cancel the greatest common factor:

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

3. Graph

Each line represents the function of one side of the equation:
$y = | 2 u - 6 |$
$y = | 2 u |$
The equation is true where the two lines cross.

How did we do?

Please leave us feedback.

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. 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. Graph

Why learn this

Terms and topics

Related links

Latest Related Drills Solved