Inequalities
The beauty of inequalities on the SAT is that they allow you to do all of the same things that you can with equations:
2x + 5 > 7
2x > 2 subtract 5 from both sides
x > 1. divide both sides by 2
The only - although incredibly significant - difference is that you must flip the sign of the inequality whenever you multiply or divide by a negative! Inequalities can be manipulated like equations and follow very similar rules, but there is one important exception.
If you add/subtract/multiply the same number to both sides of an inequality, the inequality remains true. But if you multiply or divide both sides of an inequality by a negative number, the inequality becomes reversed.
This is quite easy to see because we can write that 4 > 2. But if we multiply both sides of this inequality by −1, we get −4 > −2, which is not true. We have to reverse the inequality, writing as −4 < −2 in order for it to be true.
This leads to difficulties when dealing with variables because a variable can be either positive or negative.
Combining Inequalities
Many SAT inequality problems involve more than one inequality. To solve such problems, you may need to convert several inequalities to a compound inequality, which is a series of inequalities strung together, such as 2 < 3 < 4.
To convert multiple inequalities to a compound inequality, first line up the variables, then combine.
Example: If x > 8 , x < 17, and x + 5 < 19, what is the range of possible values for x.
First, solve any inequalities that need to be solved. In this example, only the last inequality needs to be solved:
x + 5 < 19
x < 14
Second, simplify the inequalities so that all the inequality symbols point in the same direction:
8 < x
x < 17
x < 14
Third, line up the common variables in the inequalities.
8 < x
x < 17
x < 14
Notice that x < 14 is more limiting than x < 17 (in other words, whenever x < l4, x will always be less than 17, but not vice versa.). That is why you choose 8 < x < 14 rather than 8 < x < 17 as the compound inequality that solves the problem. The correct answer is 8 < x < 14
Sometimes a problem with compound inequalities will require you to manipulate the inequalities in order to solve the problem. You can perform operations on a compound inequality as long as you remember to perform those operations on every term in the inequality, not just the outside terms.
Solved Example 1
If 1 > 1 - ab > 0 which of the following must be true?
I. \(\frac{a}{b}>0\)
II. \(\frac{a}{b}
III. ab < 1
A. I only
B. II only
C. I and II only
D. I and III only
Solution: You can manipulate the original compound inequality as follows, making sure to perform each manipulation on every term:
1 > 1 - ab > 0
0 > -ab > -1. Multiply all three terms by -1
0 < ab < 1. and flip the inequality signs.
Therefore you know that 0 < ab < 1. This tells you that ab is positive, so must \(\frac{a}{b}\) be positive (a and b have the same sign). Therefore, it must be true. However, you do not know whether \(\frac{a}{b}. Therefore, the correct answer is D.
Optimization Problems
Related to extreme values are problems involving optimization, specifically, minimization or maximization problems. In these problems, you need to focus on the largest and smallest possible values for each of the variables, as some combination of them will usually lead to the largest or smallest possible result.
Solved Example 2
If 2y + 3 < 11 and 1 < x < 5, what is the maximum possible value for xy?
Solution: You need to test the extreme values for x and for y to determine which combinations of extreme values will maximize xy:
- Based on 1 < x < 5, the lowest the lowest value for x is 1 and the highest value for x is 5.
- For Y, there is no lower limit to y and the highest value for y is 4.
Now consider the extreme value scenarios for X, Y, and XY:=
1. Since Y has no lower limit and x is positive, the product XY has no lower limit.
2. Using Y’s highest value (4), test the extreme values of x(1 and 5).
3. The first extreme value generates a product XY = (1)(4) = 4. The second extreme value generates XY = (5)(4) = 20.
Clearly, XY is maximized when X = 5 and Y = 4, with a result that XY = 20.
Graphing Linear Inequalities
Graphing linear inequalities is similar to graphing linear equations, but the process involves two major differences. The first difference is with regard to the line itself, depending on whether it is an inclusive or non-inclusive inequality.
- For inequalities that are inclusive (≤ or ≥), a solid line is used.
- For inequalities that are non-inclusive (< or >), a dashed line is used.
- The next aspect of graphing an inequality involves shading a region.
- If an inequality reads y is less than or y is less than or equal to, the area below the line will be shaded.
- If an inequality reads y is greater than or y is greater than or equal to, the area above the line will be shaded.
If you’re ever unsure about which side of inequality to shade, you can always choose a point to test. Consider the inequality y > x − 4. Let’s say that we’re unsure about which side of the inequality to shade. We can choose the value of (0,0) and plug it into the equation:
0 > 0 − 4
0 > − 4
With these values inserted, the inequality holds true, so we want to shade the side of the line on which the origin (0,0) falls. Any point can function as your test point; the origin is just an example. The following are graphs of linear inequalities.
System of Inequalities
Systems of inequalities are similar to systems of equations with regard to the algebraic operations needed to solve them, with the exception of the multiplication and division of negative numbers. The main difference occurs when one arrives at the solution – specifically how it is seen in the final equation and the corresponding graph. Instead of having a specific value for each variable in the system, a solution to a system of inequalities has a range of values for each variable.
