Introduction to Inequality Word Problems
Still more astonishing is that world of rigorous fantasy we call mathematics.
—GREGORY BATESON (1904–1980)
This lesson details the basics of translating and solving inequalities and word problems with inequalities. Use the information in Lesson 1 for additional review of the basics of translating words into symbols.
Translating from Words to Inequalities
The key words and phrases are summarized in the following chart.
Read through the following examples for more help with translating inequalities. Let x = a number in each example. The key phrases are in italics.
Examples
1.  Two is less than a number  2 < x 
2.  Four more than a number is greater than or equal to five.  x + 4 ≥ 5 
3.  The maximum value of a number is 28.  x ≤ 28 
4.  The sum of a number and nine is at least 81.  x + 9 ≥ 81 
Solving Inequalities
Solve inequalities by using the same golden rule as equation solving: Whatever you do to one side of the inequality, do to the other side.
Example 1
Solve the inequality for x:
 4x > 16
 Divide each side by 4.
 x > 4
Tip:When you are dividing or multiplying each side of an inequality by a negative value, the inequality symbol must switch directions. For example, to solve the inequality –5x > 10, divide each side by –5. The inequality becomes x < –2. 
Example 2
Solve the inequality for x:
+ 4 ≤ 8 + 4 – 4 ≤ 8 – 4 Subtract 4 from each side. ≤ 4 Simplify. × –2 ≥ 4 × –2 Multiply each side by –2. Switch the inequality symbol. x = –8 Simplify.
Inequality Word Problems
To solve inequality word problems, first translate from words into mathematical symbols. Then, solve the inequality using the same rules as equation solving with two differences. The solution will not be a single value, but usually a set of values. In addition, if you multiplied or divided each side of the equation by a negative number, you must switch the inequality symbol to the other direction.

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