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Solution - Absolute value equations

Exact form:

a = 4, \frac{2}{3}

a=4 , \frac{2}{3}

Decimal form:

a = 4, 0.667

a=4 , 0.667

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

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

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:

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

2. Solve the two equations for a

10 additional steps

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

Subtract from both sides:

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

Group like terms:

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

Simplify the arithmetic:

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

Group like terms:

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

Simplify the arithmetic:

$- a + 1 = - 3$

Subtract from both sides:

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

Simplify the arithmetic:

$- a = - 3 - 1$

Simplify the arithmetic:

$- a = - 4$

Multiply both sides by :

$- a \cdot - 1 = - 4 \cdot - 1$

Remove the one(s):

$a = - 4 \cdot - 1$

Simplify the arithmetic:

$a = 4$

10 additional steps

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

Expand the parentheses:

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

Add to both sides:

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

Group like terms:

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

Simplify the arithmetic:

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

Group like terms:

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

Simplify the arithmetic:

$3 a + 1 = 3$

Subtract from both sides:

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

Simplify the arithmetic:

$3 a = 3 - 1$

Simplify the arithmetic:

$3 a = 2$

Divide both sides by :

$\frac{(3 a)}{3} = \frac{2}{3}$

Simplify the fraction:

$a = \frac{2}{3}$

3. List the solutions

$a = 4, \frac{2}{3}$
(2 solution(s))

4. Graph

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