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Solution - Absolute value equations

Exact form:

a = - 2, \frac{12}{11}

a=-2 , \frac{12}{11}

Mixed number form:

a = - 2, 1 \frac{1}{11}

a=-2 , 1\frac{1}{11}

Decimal form:

a = - 2, 1.091

a=-2 , 1.091

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
$| 5 a - 7 | = | 6 a - 5 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| 5 a - 7 \| = \| 6 a - 5 \|$
$x = + y$	$(5 a - 7) = (6 a - 5)$
$x = - y$	$(5 a - 7) = - (6 a - 5)$
$+ x = y$	$(5 a - 7) = (6 a - 5)$
$- x = y$	$- (5 a - 7) = (6 a - 5)$

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:

$\| x \| = \| y \|$	$\| 5 a - 7 \| = \| 6 a - 5 \|$
$x = + y$ , $+ x = y$	$(5 a - 7) = (6 a - 5)$
$x = - y$ , $- x = y$	$(5 a - 7) = - (6 a - 5)$

2. Solve the two equations for a

10 additional steps

$(5 a - 7) = (6 a - 5)$

Subtract from both sides:

$(5 a - 7) - 6 a = (6 a - 5) - 6 a$

Group like terms:

$(5 a - 6 a) - 7 = (6 a - 5) - 6 a$

Simplify the arithmetic:

$- a - 7 = (6 a - 5) - 6 a$

Group like terms:

$- a - 7 = (6 a - 6 a) - 5$

Simplify the arithmetic:

$- a - 7 = - 5$

Add to both sides:

$(- a - 7) + 7 = - 5 + 7$

Simplify the arithmetic:

$- a = - 5 + 7$

Simplify the arithmetic:

$- a = 2$

Multiply both sides by :

$- a \cdot - 1 = 2 \cdot - 1$

Remove the one(s):

$a = 2 \cdot - 1$

Simplify the arithmetic:

$a = - 2$

10 additional steps

$(5 a - 7) = - (6 a - 5)$

Expand the parentheses:

$(5 a - 7) = - 6 a + 5$

Add to both sides:

$(5 a - 7) + 6 a = (- 6 a + 5) + 6 a$

Group like terms:

$(5 a + 6 a) - 7 = (- 6 a + 5) + 6 a$

Simplify the arithmetic:

$11 a - 7 = (- 6 a + 5) + 6 a$

Group like terms:

$11 a - 7 = (- 6 a + 6 a) + 5$

Simplify the arithmetic:

$11 a - 7 = 5$

Add to both sides:

$(11 a - 7) + 7 = 5 + 7$

Simplify the arithmetic:

$11 a = 5 + 7$

Simplify the arithmetic:

$11 a = 12$

Divide both sides by :

$\frac{(11 a)}{11} = \frac{12}{11}$

Simplify the fraction:

$a = \frac{12}{11}$

3. List the solutions

$a = - 2, \frac{12}{11}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | 5 a - 7 |$
$y = | 6 a - 5 |$
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 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

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