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Test PrepSATMathInequalities

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

  1. Solve the inequality 3x + 5 < 5x - 9

    (A) x > 7
    (B) x < 7
    (C) x > 2
    (D) x < 2
    (E) x = 7

  2. 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

  1. 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 test-writers also give you x < 7 as an answer choice, so don’t forget to switch the inequality sign!
  2. 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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