Math Study Guide for the SAT Exam

Inequalities

Inequalities are math statements compare two unequal quantities. Working with inequalities is similar to working with equations, but there are some differences. Here are some things to know.

Linear Inequalities in One or Two Variables

A linear inequality compares \(x\) and \(y\) expressions using one of four symbols: greater than (\(\gt\)), greater than or equal to (\(\ge\)), less than (\(\lt\)), or less than or equal to (\(\le\)). With linear inequalities, instead of finding a distinct point of intersection between two lines, the solution set will be a range of values that satisfy the stated parameters.

Linear inequalities are simplified in the same way that linear equations are simplified, with one major difference. When a linear inequality is multiplied or divided by a negative integer, the direction of the inequality sign switches. For example:

\[-4x < 8\] \[x > -2\]

One Variable

As with linear equations, single-variable linear inequalities are solved by manipulating the inequality so that the variable is on one side and a value is on the other. Consider the following linear inequality:

\[3 + 2(4 - x) \le 4x - 1\]

Let’s solve this inequality by first distributing the \(2\):

\[3 + 8 - 2x \le 4x - 1\]

Next, we’ll combine like terms:

\[11 - 2x \le 4x - 1\]

Now, let’s rearrange the terms:

\[12 \le 6x\]

Finally, we divide both sides by \(6\):

\[2 \le x\]

This solution can be verified by selecting a value for \(x\) that is greater than or equal to \(2\) and testing if it yields a true statement. We will leave the verification to you.

Two Variables

As with a linear equation with two variables, a two-variable linear inequality is solved in terms of one of the variables. Consider the following linear inequality:

\[-3x - 4(y + 3) > 2y + 2(3 - x)\]

We will begin by manipulating the inequality to solve for \(y\) in terms of \(x\):

First, we’ll distribute the \(4\):

\[-3x - 4y - 12 > 2y + 6 - 2x\]

We can now combine like terms through rearrangement:

\[-4y - 2y > -2x + 3x + 6 + 12\] \[-6y > x + 18\]

Recall, when multiplying or dividing both sides of an inequality by a negative value, the inequality sign switches directions:

\[\frac{-6y}{-6} < \frac{x + 18}{-6}\] \[y < \frac{-1}{6}x - 3\]

Graphing Inequalities

Inequalities are graphed differently than equations. We aren’t just graphing a line; our graph must illustrate the full solution set. Consider the following situations:

\[y > x\]

This inequality graphs as an upward-sloping, dotted line. The solution set is every point above the line, not including the points lying on the line (\(y=x\)).

\[y < x\]

This equation graphs similarly, except the solution set is every point below the dotted line, not including the points lying on the line.

\[y \geq x\]

This one graphs like the first, except instead of a dotted line, a bold line is used. This indicates that the points lying along the line are included in the solution set.

\[y \leq x\]

Again, this graph includes a bold line, and the area below it is shaded to indicate that every point on and below the line is included in the solution set.