# Algebra II: Solving Equations and Inequalities

On this page we hope to clear up problems that you might have with solving equations and/or inequalities.  Equations are the most common thing you'll see in all of math!  Read on or follow any of the links below to begin understanding equations and inequalities.

Dealing with fractions or decimals
Inequalities
Quiz on solving equations and inequalities

To solve an equation, you isolate the variable are solving for.  The Addition Principle says that when a = b, a + c = b + c for any number c.

Example:

```1.    Solve: x + 6 = -15
each side of the equation.

x + 6 -6 = -15 - 6

The variable is now isolated.

x = -21
```
Along the same lines, the Multiplication Principle says that if a = b and c is any number, a * c = b * c.  This principle is also used to help isolate the variable you are asked to solve for.

Example:

```2.    Solve: 4x = 9
Solution: Using the Multiplication Principle,
multiply each side of the equation by (1/4).

(1/4)4x = (1/4)9

The variable is now isolated.

x = (9/4)
```
Also, be aware of problems where you might need to use both of these principles together!

Example:

```3.    Solve: 3x - 4 = 13
each side.

3x - 4 + 4 = 13 + 4

After simplifying, 3x = 17.

Use the Multiplication Principle to multiply each
side by (1/3).

After simplification, the variable is isolated.

x = (17/3)
```

## Dealing with Fractions or Decimals

When an equation contains fractions or decimals, it usually makes it easier to solve them when the fractions or decimals aren't there.  The Multiplication Principle is used to do this.

Example:

```1.    Solve: (3/4)x + (1/2) = (3/2)
Solution: Multiply both sides by the LCM of the
denominators, 4 in this case.

Use the Distributive Law, which says a(b + c) = ab + ac
to make the equation easier to deal with.

(4 * (3/4)x) + (4 * (1/2)) = 4(3/2)

After simplification, you get an equation with no
fractions!

3x + 2 = 6

(It's left up to you to solve for x.)
```
If a and b are real numbers, and ab = 0, either a, b, or both equal 0.  This principle, called The Principle of Zero Products, is useful when you have an equation to solve that has two instances of a variable, such as (x + 3)(x - 2) = 0.

Example:

```2.    Solve: 7x(4x + 2) = 0
Solution: Using the Principle of Zero Products,

7x = 0 and 4x + 2 = 0

Solve each equation for x.

x = 0 and x = -(1/2)

The solutions are 0 and -(1/2).
```

## Inequalities

Math problems containing <, >, <=, and >= are called inequalities.  A solution to any inequality is any number that makes the inequality true.

On many occasions, you will be asked to show the solution to an inequality by graphing it on a number line.  This is usually covered in elementary algebra (Algebra I) courses.  This custom has been followed on this site, so follow this link to understand graphing on a number line.

As with equations, inequalities also have principles dealing with addition and multiplication.  They are outlined below.

1Addition Principle for Inequality — If a > b then a + c > b + c.

Example:

```1.    Solve: x + 3 > 6
each side of the inequality.

x + 3 - 3 > 6 - 3

After simplification, x > 3.
```
2Multiplication Principle for Inequalities — If a > b and c is positive, then ac > bc.  If a > b and c is negative, then ac < bc (notice the sign was reversed).

Example:

```1.    Solve: -4x < .8
Solution: Using the Multiplication Principle, multiply both
sides of the inequality by -.25.  Then reverse the
signs.

-.25(-4x) > -.25(.8)

x > -.2
```
One thing in math that seems to give people trouble throughout their math careers is absolute value.  The absolute value of any number is its numerical value (ignore the sign).  For example, the absolute value of -6 is 6 and |+3| (the vertical lines stand for absolute value) is 3.

Absolute value in inequalities is a little more complicated.  For example, |x| >= 4 asks us for all numbers that have an absolute value that is greater than or equal to 4.  Obviously, 4 and any number greater than 4 is a solution.  The confusing part comes from the fact that -4 and any number less than -4 is a solution (|-4| = 4, |-5| = 5, etc.).  Therefore, the solution is x >= 4 or x <= -4.

Absolute value becomes even more complicated when dealing with equations.  However, there is a theorem that tells us how to deal with equations with absolute value and complicated inequalities.

1.  If X is any expression, and b any positive number, and |X| = b it is the same as |X| = b or |X| = -b.

2.  If X is any expression, and b any positive number, and |X| < b it is the same as -b < X < b.

3.  If X is any expression, and b any positive number, and |X| > b it is the same as X < -b, X > b.

Example:

```1.    Solve: |5x - 4| = 11
Solution: Use the theorem stated above to rewrite the equation.

|X| = b
X = 5x - 4 and b = 11

5x - 4 = 11, 5x - 4 = -11

Solve each equation using the Addition Principle and the
Multiplication Principle.

5x = 15, 5x = -7
x = 3, x = -(7/5)
```

Take the quiz on solving equations and inequalities.  The quiz is very useful for either review or to see if you've really got the topic down.

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