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Solution - Absolute value equations

Exact form:

a = 0, - \frac{6}{5}

a=0 , -\frac{6}{5}

Mixed number form:

a = 0, - 1 \frac{1}{5}

a=0 , -1\frac{1}{5}

Decimal form:

a = 0, - 1.2

a=0 , -1.2

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
$| 2 a + 3 | = 3 | a + 1 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| 2 a + 3 \| = 3 \| a + 1 \|$
$x = + y$	$(2 a + 3) = 3 (a + 1)$
$x = - y$	$(2 a + 3) = 3 (- (a + 1))$
$+ x = y$	$(2 a + 3) = 3 (a + 1)$
$- x = y$	$- (2 a + 3) = 3 (a + 1)$

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:

$\| x \| = \| y \|$	$\| 2 a + 3 \| = 3 \| a + 1 \|$
$x = + y$ , $+ x = y$	$(2 a + 3) = 3 (a + 1)$
$x = - y$ , $- x = y$	$(2 a + 3) = 3 (- (a + 1))$

2. Solve the two equations for a

12 additional steps

$(2 a + 3) = 3 \cdot (a + 1)$

Expand the parentheses:

$(2 a + 3) = 3 a + 3 \cdot 1$

Simplify the arithmetic:

$(2 a + 3) = 3 a + 3$

Subtract from both sides:

$(2 a + 3) - 3 a = (3 a + 3) - 3 a$

Group like terms:

$(2 a - 3 a) + 3 = (3 a + 3) - 3 a$

Simplify the arithmetic:

$- a + 3 = (3 a + 3) - 3 a$

Group like terms:

$- a + 3 = (3 a - 3 a) + 3$

Simplify the arithmetic:

$- a + 3 = 3$

Subtract from both sides:

$(- a + 3) - 3 = 3 - 3$

Simplify the arithmetic:

$- a = 3 - 3$

Simplify the arithmetic:

$- a = 0$

Multiply both sides by :

$- a \cdot - 1 = 0 \cdot - 1$

Remove the one(s):

$a = 0 \cdot - 1$

Multiply by zero:

$a = 0$

14 additional steps

$(2 a + 3) = 3 \cdot (- (a + 1))$

Expand the parentheses:

$(2 a + 3) = 3 \cdot (- a - 1)$

$(2 a + 3) = 3 \cdot - a + 3 \cdot - 1$

Group like terms:

$(2 a + 3) = (3 \cdot - 1) a + 3 \cdot - 1$

Multiply the coefficients:

$(2 a + 3) = - 3 a + 3 \cdot - 1$

Simplify the arithmetic:

$(2 a + 3) = - 3 a - 3$

Add to both sides:

$(2 a + 3) + 3 a = (- 3 a - 3) + 3 a$

Group like terms:

$(2 a + 3 a) + 3 = (- 3 a - 3) + 3 a$

Simplify the arithmetic:

$5 a + 3 = (- 3 a - 3) + 3 a$

Group like terms:

$5 a + 3 = (- 3 a + 3 a) - 3$

Simplify the arithmetic:

$5 a + 3 = - 3$

Subtract from both sides:

$(5 a + 3) - 3 = - 3 - 3$

Simplify the arithmetic:

$5 a = - 3 - 3$

Simplify the arithmetic:

$5 a = - 6$

Divide both sides by :

$\frac{(5 a)}{5} = \frac{- 6}{5}$

Simplify the fraction:

$a = \frac{- 6}{5}$

3. List the solutions

$a = 0, - \frac{6}{5}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | 2 a + 3 |$
$y = 3 | a + 1 |$
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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