Inequalities Study Guide
Introduction to Inequalities
Numbers are the highest degree of knowledge. It is knowledge itself.
—Plato (424–348 B.C.)
In this lesson, you will learn what an inequality is and how to solve for the variable in an inequality.
Here's the scenario. You're saving up money for an MP3 player that costs $130. You've been pooling your allowance, part-time job pay, and monetary gifts for the past few months. You know you need at least $130. When you have saved $130 or more, you'll be able to get your MP3 player.
Question: How many numbers satisfy this requirement? In other words, how many amounts are more than $130? Your solution set would include $130 and amounts larger than $130. A question such as this can be solved using inequalities, instead of equations. It would be written as follows:
- r ≥ $130
Inequalities are sentences that compare quantities. Inequalities contain the greater than, less than, greater than or equal to (as shown in the example), or less than or equal to symbols. The following chart visually shows the inequality symbols, as well as their verbal equivalents.
When you solve the inequalities for a variable, you can figure out a range of numbers that your unknown is allowed to be. Here are a few examples of inequalities:
- 17 > 12 is read "seventeen is greater than twelve."
- –61 < –35 is read "negative sixty-one is less than negative thirty-five."
On a number line, > and < are represented by an empty circle at the end of a line segment on a number graph. When you use the symbol < or >, make an open circle at this number to show this number is not part of the solution set.
On a number line, ≥ and ≤ are represented by a solid circle at the end of a line segment on a number line graph. When you use the symbol ≤ or ≥, place a closed, or filled-in, circle at this number to show this number is a part of the solution set.
For example, the graph of the inequality –7 ≤ x < 5 would look like this:
Solving an inequality means finding all of its solutions. A solution of an inequality is a number that can be substituted for a variable and makes the inequality true.
Changing an Inequality
You can change an inequality in the same way you can change an equation—you can add the same number to both sides, subtract the same number, and so on. There is, however, one rule that you need to remember when you deal with inequalities:
When you multiply or divide by a negative number, you need to reverse the inequality sign.
If x > y, then –x < –y. Let's see how this works: –5x + 3 > 28 can also be expressed as which of the following? Remember, the goal is to isolate the variable x.
Okay, time to apply the rule. When you multiply or divide by a negative number, you need to reverse the sign. So when you divide by –5, you get:
x < –5
To graph a solution set on a number line, use the number in the solution as the starting point on the number line. In the problem x < –5, the starting point on the number line is –5. Because the symbol < is used, make an open circle at this number to show this number is not part of the solution set. Next, draw an arrow from that point to the left on the number line, because solutions to this problem are less than –5.
Let's try another one. Solve for x:
12 – 6x > 0
First, you need to subtract 12 from both sides.
Now, divide both sides by –6. Notice that the sign flips this time, because you are dividing by a negative number.
- x < 2
To graph this solution set on a number line, 2 is the starting point. Because the symbol < is used, make an open circle and draw an arrow from that point to the left on the number line.
You could solve for x in this inequality without ever multiplying or dividing by a negative number. How? Just add 6x to both sides and the sign stays the same. Then, divide both sides by 6.
Solving Compound Inequalities
A compound inequality is a combination of two or more inequalities, such as –3 < x + 1 < 4. How would you solve this?
Start by subtracting 1 from all parts of the inequality to get the variable by itself:
–3 – 1 < x + 1 – 1 < 4 – 1 –4 < x < 3
The solution set for this compound inequality is all numbers between –4 and 3. On a number line, the solution set looks like this:
Find practice problems and solutions for these concepts at Inequalities Practice Questions.
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