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# Algebraic Inequalities Study Guide (page 2)

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Updated on Oct 3, 2011

## Compound Inequalities

So far, we have seen inequalities where a variable is less than, less than or equal to, greater than, or greater than or equal to a quantity. Sometimes, a variable is between two quantities. For example, if x can be as small as –4 but less than 7, we would write –4 ≤ x < 7. The number –4 is part of the solution because x can be –4, but 7 is not part of the solution because x is less than 7. We can show this inequality on a number line, too. No arrows are highlighted, because the solution set has two boundaries:

An equation and a simple inequality both have left and right sides. Whatever operation we perform on one side, we perform on the other. A compound inequality has three parts, since it has two inequality symbols. Whatever operation we perform on one part, we must perform on all three parts.

To simplify the compound inequality 7 < x + 2 < 9, isolate the variable, x, in the center of the inequality. Do this by subtracting 2 from all three parts of the inequality, which gives us 5 < x < 7.

To simplify the inequality –4 ≤ –2x ≤ 10, we must divide each part of the inequality by –2. Because we are dividing by a negative number, we must change both inequality symbols: –4 ≤ –2x ≤ 10 becomes 2 ≥ x ≥ –5.

## Inequalities With Two Variables

When we are given an inequality with both x and y, we simplify it by writing y in terms of x.

Given the inequality 12x + 3y < 6, we subtract 12x from both sides and divide by 3:

12x + 3y < 6
3y < 6 – 12x
y < 2 – 4x

Just as with single-variable inequalities, if we multiply or divide both sides of the inequality by a negative number, we must change the inequality symbol. To solve x – 5y ≥ 15 for y in terms of x, we subtract x from both sides, divide by –5, and change the inequality symbol from ≥ to ≤ :

x – 5y ≥ 15
–5y ≥ 15 – x

Find practice problems and solutions for these concepts at Algebraic Inequalities Practice Questions.

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