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Solution - Absolute value equations

Exact form:

a = 3, 1

a=3 , 1

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
$| 3 a - 4 | = | 2 a - 1 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| 3 a - 4 \| = \| 2 a - 1 \|$
$x = + y$	$(3 a - 4) = (2 a - 1)$
$x = - y$	$(3 a - 4) = - (2 a - 1)$
$+ x = y$	$(3 a - 4) = (2 a - 1)$
$- x = y$	$- (3 a - 4) = (2 a - 1)$

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:

$\| x \| = \| y \|$	$\| 3 a - 4 \| = \| 2 a - 1 \|$
$x = + y$ , $+ x = y$	$(3 a - 4) = (2 a - 1)$
$x = - y$ , $- x = y$	$(3 a - 4) = - (2 a - 1)$

2. Solve the two equations for a

7 additional steps

$(3 a - 4) = (2 a - 1)$

Subtract from both sides:

$(3 a - 4) - 2 a = (2 a - 1) - 2 a$

Group like terms:

$(3 a - 2 a) - 4 = (2 a - 1) - 2 a$

Simplify the arithmetic:

$a - 4 = (2 a - 1) - 2 a$

Group like terms:

$a - 4 = (2 a - 2 a) - 1$

Simplify the arithmetic:

$a - 4 = - 1$

Add to both sides:

$(a - 4) + 4 = - 1 + 4$

Simplify the arithmetic:

$a = - 1 + 4$

Simplify the arithmetic:

$a = 3$

11 additional steps

$(3 a - 4) = - (2 a - 1)$

Expand the parentheses:

$(3 a - 4) = - 2 a + 1$

Add to both sides:

$(3 a - 4) + 2 a = (- 2 a + 1) + 2 a$

Group like terms:

$(3 a + 2 a) - 4 = (- 2 a + 1) + 2 a$

Simplify the arithmetic:

$5 a - 4 = (- 2 a + 1) + 2 a$

Group like terms:

$5 a - 4 = (- 2 a + 2 a) + 1$

Simplify the arithmetic:

$5 a - 4 = 1$

Add to both sides:

$(5 a - 4) + 4 = 1 + 4$

Simplify the arithmetic:

$5 a = 1 + 4$

Simplify the arithmetic:

$5 a = 5$

Divide both sides by :

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

Simplify the fraction:

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

Simplify the fraction:

$a = 1$

3. List the solutions

$a = 3, 1$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | 3 a - 4 |$
$y = | 2 a - 1 |$
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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