SAT Math Skill Review: Inequalities
An inequality is a statement indicating that one quantity is greater than or less than another. Inequalities are shown using four symbols:
Symbol

Meaning

Example

>

Greater than


≥

Greater than or equal to


<

Less than


≤

Less than or equal to


Solving inequalities works just like solving an equality (an equation with an “=” sign), except for one important difference:
When multiplying or dividing an inequality by a negative number, you must switch the sign.
Let’s take a look at an example:
Your first step is to isolate the variable. To do this, add 5 to each side:
Next, divide both sides by 2. Because 2 is a positive number, you don’t need to worry about switching the signs:
This expression means that all values of x that are greater than 7 are solutions to this inequality.
Examples
 Solve the inequality 3x + 5 < 5x  9
(A) x > 7
(B) x < 7
(C) x > 2
(D) x < 2
(E) x = 7

Which of the following algebraic inequalities is represented by the graph below?
(A) 2 ≥ y < 2
(B) 2 < y > 2
(C) 2 ≤ y < 2
(D) 2 < y ≤ 2
(E) 2 ≤ y ≤ 2
Answers and Explanations
 The correct answer is A. First, subtract 5 from both sides to get 3x < 5x – 14. Then, subtract 5x from both sides to get 2x < 14. To solve for x, divide both sides by 2 and, since you are dividing by a negative number, switch the inequality sign to get your final solution: x > 7. Notice that the testwriters also give you x < 7 as an answer choice, so don’t forget to switch the inequality sign!
 The correct answer is C. The graph represents a line whose solution y lies between 2 and 2. The filled point represents a true value for y. Based on the graph, y = 2. This allows you to eliminate choices B and D which don’t allow for this value. The open point represents y ≠ 2. This allows you to eliminate choices D and E. To answer the question, you must combine the two inequalities into one statement: 2 ≤ y < 2. Only choices A and C remain to check.
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