SAT Tip: The Golden Rule of Inequalities

Learning how to solve for inequalities is a must when prepping for the Math section of the SAT. Photograph by STOCK4B

This tip on improving your SAT score was provided by Vivian Kerr at Veritas Prep.

Inequalities are an important SAT algebra concept, and you will definitely see them on the SAT Math section. Often this concept is combined with absolute value. The symbols for inequalities are (“greater than”), ≤ (“less than or equal to”), and ≥ (“greater than or equal to”). For example, x ≤ 1 means that x can be any number less than 1, or it can be 1 itself. You’ve probably seen questions with the words inclusive or exclusive. This is the same idea. x ≤ 1 means including 1, while x

It can be helpful to think of the inequality sign like the mouth of a crocodile. The crocodile always wants to eat the bigger number (or person!). The inequality will always open towards the larger number.  What makes inequalities special is the fact that they represent a range of possible solutions. Unlike an equation, there are many correct answers for x. When you solve an inequality, what you are left with after simplifying is simply an expression for the possible range of x.

When you solve an inequality, you solve exactly the same way you do as an equation. Just pretend the inequality sign is an equal sign! For example:

5x + 3 > 18

Subtracting 3 from each side of the equation results in:

5x > 15

Dividing each side of the equation by 5 results in:

x > 3

The only difference between an inequality and an equation is that if this had been an equation, our answer would have been x = 3. For this inequality, x can equal any value greater than 3, but not 3 itself.

Now to graph this inequality on a number line:

Remember that on a number line, the numbers get larger as we move to the right, and smaller as we move to the left. Since our range of possible values is all numbers greater than 3, we move to the right. That is why the arrow points right. The circle is open since x ≠ 3. If x ≥ 3, we would have bubbled in the circle.

Finally, there is one all-important “Golden Rule” to keep in mind when solving inequalities: When you multiply or divide by a negative, you must reverse the direction of the inequality. For example:

- 2x + 6 ≤ 16

Subtracting 6 from both sides of the equation results in:

- 2x ≤ 10

Dividing both sides by -2 (here’s where “the flip” occurs!) results in:

x ≥ -5

When we divided the -2 by both sides of the inequality, we had to reverse the sign since -2 is not positive. If we were multiplying by -2, we would have done the same thing. The SAT is notorious for offering incorrect answer choices with the sign reversed.  For a question like this, you can bet x ≥ -5 and x ≤ -5 would both have been answer options.  Remember this “Golden Rule” to help you think more like the test-maker.

Vivian Kerr has been teaching and tutoring in the Los Angeles area since 2005. She graduated from the University of Southern California, studied abroad in London, and has worked for several test-prep giants tutoring, writing content, and blogging about all things SAT, ACT, GRE, and GMAT.

For more SAT advice from Veritas Prep, watch “How to Cut Down the Cost of College”

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