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Solution - Absolute value equations

Exact form:

a = \frac{4}{7}, 8

a=\frac{4}{7} , 8

Decimal form:

a = 0.571, 8

a=0.571 , 8

See steps

Step-by-step explanation

1. Rewrite the equation with one absolute value terms on each side

$| 3 a + 2 | - 2 | - 2 a + 3 | = 0$

Add $2 | - 2 a + 3 |$ to both sides of the equation:

$| 3 a + 2 | - 2 | - 2 a + 3 | + 2 | - 2 a + 3 | = 2 | - 2 a + 3 |$

Simplify the arithmetic

$| 3 a + 2 | = 2 | - 2 a + 3 |$

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

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

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:

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

3. Solve the two equations for a

12 additional steps

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

Expand the parentheses:

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

Multiply the coefficients:

$(3 a + 2) = - 4 a + 2 \cdot 3$

Simplify the arithmetic:

$(3 a + 2) = - 4 a + 6$

Add to both sides:

$(3 a + 2) + 4 a = (- 4 a + 6) + 4 a$

Group like terms:

$(3 a + 4 a) + 2 = (- 4 a + 6) + 4 a$

Simplify the arithmetic:

$7 a + 2 = (- 4 a + 6) + 4 a$

Group like terms:

$7 a + 2 = (- 4 a + 4 a) + 6$

Simplify the arithmetic:

$7 a + 2 = 6$

Subtract from both sides:

$(7 a + 2) - 2 = 6 - 2$

Simplify the arithmetic:

$7 a = 6 - 2$

Simplify the arithmetic:

$7 a = 4$

Divide both sides by :

$\frac{(7 a)}{7} = \frac{4}{7}$

Simplify the fraction:

$a = \frac{4}{7}$

14 additional steps

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

Expand the parentheses:

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

Expand the parentheses:

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

Multiply the coefficients:

$(3 a + 2) = 4 a + 2 \cdot - 3$

Simplify the arithmetic:

$(3 a + 2) = 4 a - 6$

Subtract from both sides:

$(3 a + 2) - 4 a = (4 a - 6) - 4 a$

Group like terms:

$(3 a - 4 a) + 2 = (4 a - 6) - 4 a$

Simplify the arithmetic:

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

Group like terms:

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

Simplify the arithmetic:

$- a + 2 = - 6$

Subtract from both sides:

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

Simplify the arithmetic:

$- a = - 6 - 2$

Simplify the arithmetic:

$- a = - 8$

Multiply both sides by :

$- a \cdot - 1 = - 8 \cdot - 1$

Remove the one(s):

$a = - 8 \cdot - 1$

Simplify the arithmetic:

$a = 8$

4. List the solutions

$a = \frac{4}{7}, 8$
(2 solution(s))

5. Graph

Each line represents the function of one side of the equation:
$y = | 3 a + 2 |$
$y = 2 | - 2 a + 3 |$
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 with one absolute value terms on each side

2. Rewrite the equation without absolute value bars

3. Solve the two equations for a

4. List the solutions

5. Graph

Why learn this

Terms and topics

Related links

Solution - Absolute value equations

Step-by-step explanation

1. Rewrite the equation with one absolute value terms on each side

2. Rewrite the equation without absolute value bars

3. Solve the two equations for a

4. List the solutions

5. Graph

Why learn this

Terms and topics

Related links

Latest Related Drills Solved