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Solution - Absolute value equations

Exact form:

a = \frac{3}{8}

a=\frac{3}{8}

Decimal form:

a = 0.375

a=0.375

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
$| 4 a + 5 | = | - 4 a + 8 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| 4 a + 5 \| = \| - 4 a + 8 \|$
$x = + y$	$(4 a + 5) = (- 4 a + 8)$
$x = - y$	$(4 a + 5) = - (- 4 a + 8)$
$+ x = y$	$(4 a + 5) = (- 4 a + 8)$
$- x = y$	$- (4 a + 5) = (- 4 a + 8)$

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:

$\| x \| = \| y \|$	$\| 4 a + 5 \| = \| - 4 a + 8 \|$
$x = + y$ , $+ x = y$	$(4 a + 5) = (- 4 a + 8)$
$x = - y$ , $- x = y$	$(4 a + 5) = - (- 4 a + 8)$

2. Solve the two equations for a

9 additional steps

$(4 a + 5) = (- 4 a + 8)$

Add to both sides:

$(4 a + 5) + 4 a = (- 4 a + 8) + 4 a$

Group like terms:

$(4 a + 4 a) + 5 = (- 4 a + 8) + 4 a$

Simplify the arithmetic:

$8 a + 5 = (- 4 a + 8) + 4 a$

Group like terms:

$8 a + 5 = (- 4 a + 4 a) + 8$

Simplify the arithmetic:

$8 a + 5 = 8$

Subtract from both sides:

$(8 a + 5) - 5 = 8 - 5$

Simplify the arithmetic:

$8 a = 8 - 5$

Simplify the arithmetic:

$8 a = 3$

Divide both sides by :

$\frac{(8 a)}{8} = \frac{3}{8}$

Simplify the fraction:

$a = \frac{3}{8}$

6 additional steps

$(4 a + 5) = - (- 4 a + 8)$

Expand the parentheses:

$(4 a + 5) = 4 a - 8$

Subtract from both sides:

$(4 a + 5) - 4 a = (4 a - 8) - 4 a$

Group like terms:

$(4 a - 4 a) + 5 = (4 a - 8) - 4 a$

Simplify the arithmetic:

$5 = (4 a - 8) - 4 a$

Group like terms:

$5 = (4 a - 4 a) - 8$

Simplify the arithmetic:

$5 = - 8$

The statement is false:

$5 = - 8$

The equation is false so it has no solution.

3. List the solutions

$a = \frac{3}{8}$
(1 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | 4 a + 5 |$
$y = | - 4 a + 8 |$
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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