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Inequalities and Absolute Value Concepts -- Inequalities

Module by: Kenny M. Felder. E-mail the author

Summary: This module introduces the concept of inequalities.

The symbols for inequalities are familiar:

  • x<7x<7 size 12{x<7} {} x x is less than 7”
  • x>7x>7 size 12{x>7} {} x x is greater than 7”
  • x7x7 size 12{x <= 7} {} x x is less than or equal to 7”
  • x7x7 size 12{x >= 7} {} x x is greater than or equal to 7”

If you have trouble remembering which is which, it may be helpful to remember that the larger side of the < < size 12{<} {} symbol always goes with the larger number. Hence, when you write x<7x<7 size 12{x<7} {} you can see that the 7 is the larger of the two numbers. Some people think of the < < size 12{<} {} symbol as an alligator’s mouth, which always opens toward the largest available meal!

Visually, we can represent these inequalities on a number line. An open circle is used to indicate a boundary that is not a part of the set; a closed circle is used for a boundary that is a part of the set.

Figure 1
(a) Includes all numbers less than 7, but not 7; x < 7 x < 7 ; ( - , 7 ) ( - , 7 ) (b) Includes all numbers less than 7, and 7 itself; x 7 x 7 ; ( - , 7 ] ( - , 7 ]
A number line w/ the shaded interval (-∞, 7) A number line w/ the shaded interval (-∞, 7]

AND and OR

More complicated intervals can be represented by combining these symbols with the logical operators AND and OR.

For instance, “ x3x3 size 12{x >= 3} {} AND x<6x<6 size 12{x<6} {}” indicates that x x size 12{x} {} must be both greater-than-or-equal-to 3, and less-than 6. A number only belongs in this set if it meets both conditions. Let’s try a few numbers and see if they fit.

Table 1
Sample number x 3 x 3 size 12{x >= 3} {} x < 6 x < 6 size 12{x<6} {} x3x3 size 12{x >= 3} {} AND x<6x<6 size 12{x<6} {} (both true)
x = 8 x = 8 size 12{x=8} {} Yes No No
x = 0 x = 0 size 12{x=0} {} No Yes No
x = 4 x = 4 size 12{x=4} {} Yes Yes Yes

We can see that a number must be between 3 and 6 in order to meet this AND condition.

Figure 2: x 3 AND x < 6 x 3 AND x < 6 All numbers that are greater-than-or-equal-to 3, and are also less than 6; 3 x < 6 3 x < 6
A number line w/ the shaded interval [3, 6)

This type of set is sometimes represented concisely as 3x<63x<6 size 12{3 <= x<6} {}, which visually communicates the idea that xx size 12{x} {} is between 3 and 6. This notation always indicates an AND relationship.

x<3x<3 size 12{x < 3} {} OR x6x6 size 12{x>=6} {}” is the exact opposite. It indicates that xx size 12{x} {} must be either less-than 3, or greater-than-or-equal-to 6. Meeting both conditions is OK, but it is not necessary.

Table 2
Sample number x < 3 x < 3 size 12{x < 3} {} x 6 x 6 size 12{x>6} {} x<3x<3 size 12{x < 3} {} OR x6x6 size 12{x>6} {} (either one or both true)
x = 8 x = 8 size 12{x=8} {} No Yes Yes
x = 0 x = 0 size 12{x=0} {} Yes No Yes
x = 4 x = 4 size 12{x=4} {} No No No

Visually, we can represent this set as follows:

Figure 3: All numbers that are either less than 3, or greater-than-or-equal-to 6; x < 3 OR x 6 x < 3 OR x 6
The number line with (-infinity,3) and [6,infinity] intervals shaded.

Both of the above examples are meaningful ways to represent useful sets. It is possible to put together many combinations that are perfectly logical, but are not meaningful or useful. See if you can figure out simpler ways to write each of the following conditions.

  1. x3x3 size 12{x >= 3} {} AND x>6x>6 size 12{x>6} {}
  2. x3x3 size 12{x >= 3} {} OR x>6x>6 size 12{x>6} {}
  3. x<3x<3 size 12{x<3} {} AND x>6x>6 size 12{x>6} {}
  4. x>3x>3 size 12{x>3} {} OR x<6x<6 size 12{x<6} {}

If you are not sure what these mean, try making tables of numbers like the ones I made above. Try a number below 3, a number between 3 and 6, and a number above 6. See when each condition is true. You should be able to convince yourself of the following:

  1. The first condition above is filled by any number greater than 6; it is just a big complicated way of writing x>6x>6 size 12{x>6} {}.
  2. Similarly, the second condition is the same as x3x3 size 12{x >= 3} {}.
  3. The third condition is never true.
  4. The fourth condition is always true.

I have to pause here for a brief philosophical digression. The biggest difference between a good math student, and a poor or average math student, is that the good math student works to understand things; the poor student tries to memorize rules that will lead to the right answer, without actually understanding them.

The reason this unit (Inequalities and Absolute Values) is right here at the beginning of the book is because it distinguishes sharply between these two kinds of students. Students who try to understand things will follow the previous discussion of AND and OR and will think about it until it makes sense. When approaching a new problem, they will try to make logical sense of the problem and its solution set.

But many students will attempt to learn a set of mechanical rules for solving inequalities. These students will often end up producing nonsensical answers such as the four listed above. Instead of thinking about what their answers mean, they will move forward, comfortable because “it looks sort of like the problem the teacher did on the board.”

If you have been accustomed to looking for mechanical rules to follow, now is the time to begin changing your whole approach to math. It’s not too late!!! Re-read the previous section carefully, line by line, and make sure each sentence makes sense. Then, as you work problems, think them through in the same way: not “whenever I see this kind of problem the answer is an and” but instead “What does AND mean? What does OR mean? Which one correctly describes this problem?”

All that being said, there are still a few hard-and-fast rules that I will point out as I go. These rules are useful—but they do not relieve you of the burden of thinking.

One special kind of OR is the symbol ±± size 12{ +- {}} {}. Just as size 12{>=} {} means “greater than OR equal to,” ±± size 12{ +- {}} {} means “plus OR minus.” Hence, if x2=9x2=9 size 12{x rSup { size 8{2} } =9} {}, we might say that x=±3x=±3 size 12{x= +- 3} {}; that is, xx size 12{x} {} can be either 3, or –3.

Another classic sign of “blind rule-following” is using this symbol with inequalities. What does it mean to say x<±3x<±3 size 12{x< +- 3} {}? If it means anything at all, it must mean “ x<3x<3 size 12{x<3} {} OR x<3x<3 size 12{x< - 3} {}”; which, as we have already seen, is just a sloppy shorthand for x<3x<3 size 12{x<3} {}. If you find yourself using an inequality with a ±± size 12{ +- {}} {} sign, go back to think again about the problem.

Hard and fast rule:

Inequalities and the ±± size 12{ +- {}} {} symbol don’t mix.

Solving Inequalities

Inequalities are solved just like equations, with one key exception.

Hard and fast rule:

Whenever you multiply or divide by a negative number, the sign changes.

You can see how this rule affects the solution of a typical inequality problem:

Table 3
3x + 4 > 5x + 10 3x + 4 > 5x + 10 size 12{3x+4>5x+"10"} {} An “inequality” problem
2x + 4 > 10 2x + 4 > 10 size 12{ - 2x+4>"10"} {} subtract 5 x 5 x from both sides
2x > 6 2x > 6 size 12{ - 2x>6} {} subtract 4 from both sides
x < 3 x < 3 size 12{x< - 3} {} divide both sides by –2, and change sign!

As always, being able to solve the problem is important, but even more important is knowing what the solution means. In this case, we have concluded that any number less than –3 will satisfy the original equation, 3x+4>5x+103x+4>5x+10 size 12{3x+4>5x+"10"} {}. Let’s test that.

Table 4
x=4x=4 size 12{x= - 4} {}: 3(4)+4>5(4)+103(4)+4>5(4)+10 size 12{3 \( - 4 \) +4>5 \( - 4 \) +"10"} {} 8>108>10 size 12{ - 8> - "10"} {} Yes.
x=2x=2 size 12{x= - 2} {}: 3(2)+4>5(2)+103(2)+4>5(2)+10 size 12{3 \( - 2 \) +4>5 \( - 2 \) +"10"} {} 2>02>0 size 12{ - 2>0} {} No.

As expected, x=4x=4 size 12{x= - 4} {} (which is less than –3) works; x=2x=2 size 12{x= - 2} {} (which is not) does not work.

Why do you reverse the inequality when multiplying or dividing by a negative number? Because negative numbers are backward! 5 is greater than 3, but –5 is less than –3. Multiplying or dividing by negative numbers moves you to the other side of the number line, where everything is backward.

Figure 4: Multiplying by -1 moves you "over the rainbow" to the land where everything is backward!
A number line illustration demonstrating the effects of multiplying and/or dividing an inequality by a negative number.

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