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Algebra Review Help (page 3)

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Updated on Oct 27, 2011

Inequalities

An inequality is two numbers or expressions that are connected with an inequality symbol. Inequalities contain the greater than, less than, greater than or equal to, less than or equal, or not equal to symbols. When you solve the inequalities for x, you can figure out a range of numbers that your unknown is allowed to be.

Inequalities

Here are some examples of inequalities:

2 < 5 (2 is less than 5)
9 > 3 (9 is greater than 3)
4 ≤ 4 (4 is less than or equal to 4)
2x + 5 ≠ 11 (2x added to 5 is not equal to 11)

Now, there is one rule that you need to remember when dealing with inequalities: When you multiply or divide by a negative number, you need to reverse the sign.

Tip: Whenever you multiply or divide an inequality by a negative number, you need to reverse the inequality symbol.

Example

–5x + 3 > 28 can also be expressed as what inequality?

The goal here is to isolate your x. First, subtract 3 from both sides.

–5x + 3 – 3 > 28 – 3

–5x              > 25

When you multiply or divide by a negative number, you need to reverse the sign. When you divide by –5, you get:

So, x < –5.

Keep Positive - Rearranging Inequalities to Produce Positive Coefficients

Here's a Trick to avoid having to worry about flipping the sign: Just move your terms in a manner such that you will end up with a positive coefficient on your variable.

Let's look at –5x + 3 > 28 again. Our x coefficient is negative, so you will add +5x to both sides.

–5x + 3 + 5x > 28 + 5x

        3 > 28 + 5x

Next, subtract 28 from both sides:

3 – 28 > 28 + 5x – 28

3 – 28 > 5x

   –25 > 5x

You divide both sides by 5 to yield:

–5 > x

Solving Inequalities

You can solve inequalities with variables just like you can solve equations with variables. Use what you already know about solving equations to solve inequalities. Like equations, you can add, subtract, multiply, or divide both sides of an inequality with the same number. In other words, what you do to one side of an inequality, you must do to the other side.

Example

2x + 3 < 1

Subtract 3 from both sides of the inequality. 2x + 3 – 3 < 1 – 3
Simplify both sides of the inequality. 2x < –2
Divide both sides of the inequality by 2.
Simplify both sides of the inequality. x < –1

The answer for this example is the inequality x < –1. There is an endless number of solutions because every number less than –1 is an answer. In this problem, the number –1 is not an answer because the inequality states that your answers must be numbers less than –1.

Tip: The answer to an inequality will always be an inequality. Because the answer is an inequality, you will have an infinite number of solutions.

Solving Inequalities Versus Solving Equations

Did you notice the similarity between solving equations and solving inequalities? Well, there are some major differences you need to be aware of.

Notice what happens when you multiply or divide an inequality by a negative number.

2 < 5
–2 · 2 < 5 · –2
–4 < –10

However, –4 is not less than –10. So, –4 < –10 is a false statement. To correct it, you would have to rewrite it as –4 > –10.

You can solve inequalities using the same methods that you use to solve equations with these exceptions:

  • When you multiply or divide an inequality by a negative number, you must reverse the inequality symbol.
  • The answer to an inequality will always be an inequality.

Using Variables to Express Relationships

The most important skill for solving word problems is being able to use variables to express relationships. This list will assist you in this by giving you some common examples of English phrases and their mathematical equivalents.

  • "Increase" means add.
  • "Decrease" means subtract.
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