Enter an equation or problem

Camera input is not recognized!

Solution - Absolute value equations

Exact form:

a = \frac{7}{6}, 7

a=\frac{7}{6} , 7

Mixed number form:

a = 1 \frac{1}{6}, 7

a=1\frac{1}{6} , 7

Decimal form:

a = 1.167, 7

a=1.167 , 7

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
$| - 6 a + 7 | = | 6 a - 7 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| - 6 a + 7 \| = \| 6 a - 7 \|$
$x = + y$	$(- 6 a + 7) = (6 a - 7)$
$x = - y$	$(- 6 a + 7) = - (6 a - 7)$
$+ x = y$	$(- 6 a + 7) = (6 a - 7)$
$- x = y$	$- (- 6 a + 7) = (6 a - 7)$

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:

$\| x \| = \| y \|$	$\| - 6 a + 7 \| = \| 6 a - 7 \|$
$x = + y$ , $+ x = y$	$(- 6 a + 7) = (6 a - 7)$
$x = - y$ , $- x = y$	$(- 6 a + 7) = - (6 a - 7)$

2. Solve the two equations for a

13 additional steps

$(- 6 a + 7) = (6 a - 7)$

Subtract from both sides:

$(- 6 a + 7) - 6 a = (6 a - 7) - 6 a$

Group like terms:

$(- 6 a - 6 a) + 7 = (6 a - 7) - 6 a$

Simplify the arithmetic:

$- 12 a + 7 = (6 a - 7) - 6 a$

Group like terms:

$- 12 a + 7 = (6 a - 6 a) - 7$

Simplify the arithmetic:

$- 12 a + 7 = - 7$

Subtract from both sides:

$(- 12 a + 7) - 7 = - 7 - 7$

Simplify the arithmetic:

$- 12 a = - 7 - 7$

Simplify the arithmetic:

$- 12 a = - 14$

Divide both sides by :

$\frac{(- 12 a)}{- 12} = \frac{- 14}{- 12}$

Cancel out the negatives:

$\frac{12 a}{12} = \frac{- 14}{- 12}$

Simplify the fraction:

$a = \frac{- 14}{- 12}$

Cancel out the negatives:

$a = \frac{14}{12}$

Find the greatest common factor of the numerator and denominator:

$a = \frac{(7 \cdot 2)}{(6 \cdot 2)}$

Factor out and cancel the greatest common factor:

$a = \frac{7}{6}$

5 additional steps

$(- 6 a + 7) = - (6 a - 7)$

Expand the parentheses:

$(- 6 a + 7) = - 6 a + 7$

Add to both sides:

$(- 6 a + 7) + 6 a = (- 6 a + 7) + 6 a$

Group like terms:

$(- 6 a + 6 a) + 7 = (- 6 a + 7) + 6 a$

Simplify the arithmetic:

$7 = (- 6 a + 7) + 6 a$

Group like terms:

$7 = (- 6 a + 6 a) + 7$

Simplify the arithmetic:

$7 = 7$

3. List the solutions

$a = \frac{7}{6}, 7$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | - 6 a + 7 |$
$y = | 6 a - 7 |$
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 a

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 a

3. List the solutions

4. Graph

Why learn this

Terms and topics

Related links

Latest Related Drills Solved