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# Algebra and Probability Study Guide 1 for McGraw-Hill's ASVAB (page 3)

By McGraw-Hill Professional
Updated on Jun 26, 2011

### Inequalities

Unlike equations, inequalities are statements that show that certain relationships between selected variables and numbers are not equal. Instead of using the equal sign, you use the "greater than" sign (>) or the "less than" sign (<). At times you may see signs for "greater than or equal to" (≥) or "less than or equal to" (≤).

Examples

x > 13 means that the value of x is greater than 13.

y < 45 means that the value of y is less than 45.

xy < 33 means that when y is subtracted from x, the result is less than 33.

z means that when x is divided by 5, the result is greater than or equal to the value of z.

Solving Inequalities   If you work problems with inequalities, you can treat them much like equations. If you multiply or divide both sides by a negative number, you must reverse the sign.

### Ratios and Proportions

A ratio is a comparison of one number to another. A ratio can be represented by a fraction.

A proportion is an equation stating that two ratios are equivalent. Ratios are equivalent if they can be represented by equivalent fractions. A proportion may be written

where a/b and c/d are equivalent fractions. This proportion can be read "a is to b as c is to d."

Like any other equation, a proportion can be solved for an unknown by isolating that unknown on one side of the equation. In this case, to solve for a, multiply both sides by b:

### Solving Equations for Two Unknowns

An equation may have two unknowns. An example is 3a + 3b = 9. If you are given two equations with the same unknowns, you can solve for each unknown. Here is how this process works:

Solve for a and b:
3a + 4b = 9
2a + 2b = 6

Step 1. Multiply one or both equations by a number that makes the number in front of one of the unknowns the same in both equations.

Multiply the second equation by 2 to make 4b in each equation.

2(2a + 2b = 6)
4a + 4b = 12

Step 2. Add or subtract the two equations to eliminate one unknown. Then solve for the remaining unknown.

Step 3. Insert the value for the unknown that you have found into one of the two equations. Then solve for the other unknown.

Practice questions for this study guide can be found at:

Algebra and Probability Practice Problems for McGraw-Hill's ASVAB