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Solution - Absolute value equations

Exact form:

a = \frac{3}{4}

a=\frac{3}{4}

Decimal form:

a = 0.75

a=0.75

See steps

Step-by-step explanation

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

$| 2 a - 1 | - | - 2 a + 2 | = 0$

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

$| 2 a - 1 | - | - 2 a + 2 | + | - 2 a + 2 | = | - 2 a + 2 |$

Simplify the arithmetic

$| 2 a - 1 | = | - 2 a + 2 |$

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

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

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:

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

3. Solve the two equations for a

9 additional steps

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

Add to both sides:

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

Group like terms:

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

Simplify the arithmetic:

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

Group like terms:

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

Simplify the arithmetic:

$4 a - 1 = 2$

Add to both sides:

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

Simplify the arithmetic:

$4 a = 2 + 1$

Simplify the arithmetic:

$4 a = 3$

Divide both sides by :

$\frac{(4 a)}{4} = \frac{3}{4}$

Simplify the fraction:

$a = \frac{3}{4}$

6 additional steps

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

Expand the parentheses:

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

Subtract from both sides:

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

Group like terms:

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

Simplify the arithmetic:

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

Group like terms:

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

Simplify the arithmetic:

$- 1 = - 2$

The statement is false:

$- 1 = - 2$

The equation is false so it has no solution.

4. List the solutions

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

5. Graph

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