There’s more to algebra than just equations, and inequalities will show up rather frequently on the SAT. Fortunately, many of the same strategies that apply to solving equations also apply to inequalities. There are a few key exceptions to keep in mind, but don’t worry—these will be explained throughout this chapter.
First, the language used to describe inequalities tends to be more complex than the language used to describe equations. You “solve” an equation for x, but with an inequality, you might be asked to “describe all possible values of x” or provide an answer that “includes the entire set of solutions for x.” This difference in wording exists because an equation describes a specific value of a variable, whereas an inequality describes a range of values. Regardless of the language, your task is the same: Isolate x on one side.
The following question tests your basic inequality-solving skills.
A bowling alley charges a flat $6.50 fee for shoe and ball rental plus $3.75 per game and 6.325% sales tax. If each person in a group of seven people has $20 to spend on a bowling outing, which inequality represents the maximum number of shoe and ball rentals (r) and games (g) that can be purchased by the group?
Work through the Kaplan Method for Math step-by-step to solve this question. The following table shows Kaplan’s strategic thinking on the left, along with suggested math scratchwork on the right.
| Strategic Thinking | Math Scratchwork |
|---|---|
| Step 1: Read the question, identifying and organizing important information as you go | |
| The question asks for an inequality that represents the situation described. | |
| Step 2: Choose the best strategy to answer the question | |
|
Use the Kaplan Strategy for Translating English into Math. The variables are already defined, so you only need to correctly piece them together with the given numbers. The question states that a shoe and ball rental costs $6.50 and that a game costs $3.75. Combine these with the correct variables, remembering to incorporate sales tax. Be careful when writing the right side of the inequality; the question asks for an inequality that represents the entire group, not just one person. |
r → rentals, 6.5r total $: 7 × 20 = 140 |
| Step 3: Check that you answered the right question | |
| Choice (A) contains the inequality you derived. |
Notice B is a trap answer. If you write the inequality to reflect the cost for just one person, you’ll be led right to it.
Inequalities can also be presented graphically in one or two dimensions. In one dimension, inequalities are graphed on a number line with a shaded region. For example, x > 1 could be graphed like this:
Notice the open dot at x = 1, indicating that 1 is not a solution to the inequality. This is called a strict inequality. By contrast, the graph of x ≤ 0 looks like this:
Notice the closed dot, indicating that 0 should be included in the solution set for the inequality.
In two dimensions, things get a bit more complicated. While linear equations graph as simple lines, inequalities graph as lines called boundary lines with shaded regions known as half planes. Solid lines involve inequalities that have ≤ or ≥ because the line itself is included in the solution set. Dashed lines involve strict inequalities that have > or < because, in these cases, the line itself is not included in the solution set. The shaded region represents all points that make up the solution set for the inequality.
Multiple inequalities can be combined to create a system of inequalities. This system can involve multiple variables, or it can be used to provide more detailed bounds for a range of solutions for a single variable. Systems of inequalities can also be presented graphically with multiple boundary lines and multiple shaded regions. Follow the same rules for graphing single inequalities, but keep in mind that the solution set is the region where the shading overlaps. Shading in different directions (e.g., parallel lines slanted up for one inequality and down for the other) makes the overlap easier to see. This is illustrated in the following question.
Which of the following graphs represents the solution set for 5x − 10y > 6 ?
Work through the Kaplan Method for Math step-by-step to solve this question. The following table shows Kaplan’s strategic thinking on the left, along with suggested math scratchwork on the right.
| Strategic Thinking | Math Scratchwork |
|---|---|
| Step 1: Read the question, identifying and organizing important information as you go | |
| The question is asking for the graph that matches the inequality given. | |
| Step 2: Choose the best strategy to answer the question | |
| It’s risky to eliminate choices now, as the inequality is not in slope-intercept form. Rearrange the inequality so that it’s in this form, remembering to flip the inequality symbol in the final step because you’re dividing by −10. | 
|
| The inequality in slope-intercept form indicates a negative y-intercept, so you can eliminate A and D. The “less than” symbol indicates that the shading should be below the dashed line, meaning (C) must be correct. | |
Alternatively, you can plug a point (such as the origin) into the inequality. When plugged into the inequality, you’ll see that the origin should not be in the solution set because 0 is not greater than
 . This means (C) is correct. |
|
| Step 3: Check that you answered the right question | |
| Choice (C) matches the inequality. |
Let's try solving some inequalities:
If
and
form a system of inequalities, what is one possible value of x + y ?
Work through the Kaplan Method for Math step-by-step to solve this question. The following table shows Kaplan’s strategic thinking on the left, along with suggested math scratchwork on the right.
| Strategic Thinking | Math Scratchwork |
|---|---|
| Step 1: Read the question, identifying and organizing important information as you go | |
| The question asks for a possible value of x + y. | |
| Step 2: Choose the best strategy to answer the question | |
| The question asks for an unusual quantity, x + y, so look for a shortcut. Examine the coefficients of the variables. If you write one equation under the other, you’ll see that the coefficients of the x terms sum to 1, as do the coefficients of the y terms. You can add the inequalities together (because they have the same symbol) and get x + y on one side. | 
|
| Step 3: Check that you answered the right question | |
This is a grid-in question, so pick a number between 0 and
 , inclusive, such as 0, 1, or 5/3. |
|
If 12x − 4y > 8 and
form a system of inequalities, which of the following graphs shows the solution set for the system?
Work through the Kaplan Method for Math step-by-step to solve this question. The following table shows Kaplan’s strategic thinking on the left, along with suggested math scratchwork on the right.
| Strategic Thinking | Math Scratchwork |
|---|---|
| Step 1: Read the question, identifying and organizing important information as you go | |
| You need to identify the graph that shows the solution to the system of inequalities. | |
| Step 2: Choose the best strategy to answer the question | |
| Start by rewriting each inequality in slope-intercept form. Once finished, determine what the correct graphs will look like. |
|
The boundary line for y < 3x − 2 should be a dashed line, and the boundary line for
should be a solid line. You can eliminate C based on this. The half-plane below y < 3x − 2 should be shaded, and the half-plane above
should be shaded; of the remaining choices, only (B) satisfies this requirement. |
|
| Step 3: Check that you answered the right question | |
| Choice (B) correctly depicts the solution to the system. You can check your answer by plugging a point from (B)’s solution set, such as (4, 4), into both inequalities given in the question and verifying that each results in a true statement. |