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Solution - Absolute value equations

Exact form:

y = \frac{1}{2}

y=\frac{1}{2}

Decimal form:

y = 0.5

y=0.5

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
$| 2 y + 5 | = | - 2 y + 7 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| 2 y + 5 \| = \| - 2 y + 7 \|$
$x = + y$	$(2 y + 5) = (- 2 y + 7)$
$x = - y$	$(2 y + 5) = - (- 2 y + 7)$
$+ x = y$	$(2 y + 5) = (- 2 y + 7)$
$- x = y$	$- (2 y + 5) = (- 2 y + 7)$

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:

$\| x \| = \| y \|$	$\| 2 y + 5 \| = \| - 2 y + 7 \|$
$x = + y$ , $+ x = y$	$(2 y + 5) = (- 2 y + 7)$
$x = - y$ , $- x = y$	$(2 y + 5) = - (- 2 y + 7)$

2. Solve the two equations for y

11 additional steps

$(2 y + 5) = (- 2 y + 7)$

Add to both sides:

$(2 y + 5) + 2 y = (- 2 y + 7) + 2 y$

Group like terms:

$(2 y + 2 y) + 5 = (- 2 y + 7) + 2 y$

Simplify the arithmetic:

$4 y + 5 = (- 2 y + 7) + 2 y$

Group like terms:

$4 y + 5 = (- 2 y + 2 y) + 7$

Simplify the arithmetic:

$4 y + 5 = 7$

Subtract from both sides:

$(4 y + 5) - 5 = 7 - 5$

Simplify the arithmetic:

$4 y = 7 - 5$

Simplify the arithmetic:

$4 y = 2$

Divide both sides by :

$\frac{(4 y)}{4} = \frac{2}{4}$

Simplify the fraction:

$y = \frac{2}{4}$

Find the greatest common factor of the numerator and denominator:

$y = \frac{(1 \cdot 2)}{(2 \cdot 2)}$

Factor out and cancel the greatest common factor:

$y = \frac{1}{2}$

6 additional steps

$(2 y + 5) = - (- 2 y + 7)$

Expand the parentheses:

$(2 y + 5) = 2 y - 7$

Subtract from both sides:

$(2 y + 5) - 2 y = (2 y - 7) - 2 y$

Group like terms:

$(2 y - 2 y) + 5 = (2 y - 7) - 2 y$

Simplify the arithmetic:

$5 = (2 y - 7) - 2 y$

Group like terms:

$5 = (2 y - 2 y) - 7$

Simplify the arithmetic:

$5 = - 7$

The statement is false:

$5 = - 7$

The equation is false so it has no solution.

3. List the solutions

$y = \frac{1}{2}$
(1 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | 2 y + 5 |$
$y = | - 2 y + 7 |$
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 y

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 y

3. List the solutions

4. Graph

Why learn this

Terms and topics

Related links

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