5G Reading, Understanding, Solving and Graphing Inequalities

A topic that is very important in statistics, algebra, and in many sciences is being able to use, read and understand inequalities. A linear inequality looks a lot like an equation, but uses the inequality symbols $<,\le ,>,\ge ,\text{ or}\ne$. It is vital to memorize these symbols and know what they mean. For example, statistics students routinely get P-value problems wrong not because they don’t understand P-value but because they don’t know the meaning of “ < ”. Let’s start by going over the meaning of each of these symbols and give examples using the symbols correctly.

“ < “ means “less than”. Notice the symbol looks like an arrow pointing to the left. On the number line, smaller numbers are on the left as in $-2<+10$. Notice the closed end of the inequality symbol points toward the smaller number. This is true for the four order inequality symbols. The symbol points toward the smaller number!

“ $\le$ “ means “less than or equal to”. Notice the symbol looks like the less than symbol “ < ” but now has an extra line below. “Less than or equal to” works like a “less than” symbol with the symbol pointing toward the smaller number. However “less than or equal to” now meets the added criteria that if the two numbers were equal it would still be true. For example: $-2\le +10$ is true and $+9\le +9$ is also true! (Notice that without the equal to part, $+9<+9$is not true.)

“ > “ means “greater than”. Notice the symbol looks like an arrow pointing to the right, and on the number line, larger numbers are on the right. Look at the example $+23>+7$. This is true. Notice the closed end of the inequality symbol points toward the smaller number $(+7)$ and the open end is toward the larger number $(+23)$. This is true for the four order inequality symbols. The symbol points toward the smaller number and the open end opens toward the bigger number.

“ $\ge$ “ means “greater than or equal to”. Notice the symbol looks like the “greater than” symbol “ > ” but now has an extra line below. “Greater than or equal to” works like a “greater than” symbol with the symbol pointing toward the right.
However “greater than or equal to” now meets the added criteria that if the two numbers were equal it would still be true. For example: $+17\ge -1$ is true and $-3\ge -3$ is also true! (Notice that without the equal to part, $-3>-3$is not true.)

“ $\ne$ “ means “not equal to”. Notice the symbol looks like an equal sign with a line drawn through it. This symbol is only true if the two numbers are not equal. For example $+6\ne +11$ is true but $-9\ne -9$ is not true!

Try the following inequality problems with your instructor.

Directions: Identify which number is larger and which number is smaller or if the two numbers are equal. Then determine if the symbol is used correctly or incorrectly?

Inequalities are also frequently used in algebra. For example look at the inequality . What does this mean? It is not comparing two numbers like the last examples. There is a variable involved. The most important idea to remember is the following. When we solve an equation, we are trying to find out what number or numbers we can replace the letter by so that the statement will be true. An inequality works the same way.

Ask yourself the following question. What numbers can we replace the x with that makes the inequality true? Would 4 work? If we replace x with 4 we get 4 > 3. Is that true? Yes. Then 4 is one of the solutions to the inequality $x>3$. But is 4 the only solution? Wouldn’t 3.001 or 4.5 or 5 or 10.6 or 100 or 212.37 also work? They sure would. Any number greater than 3 would work. So represents “all numbers greater than 3”.

Can we graph “all numbers greater than 3” on the number line? What would it look like?

We will draw a number line and shade all the numbers greater than 3. Be careful, it is not just whole numbers, it includes all fractions or decimals greater than 3 as well. In other words all real numbers greater than 3.

Notice that the graph starts at 3 and shades all the numbers to the right. Notice also that it begins with a parenthesis. When graphing on the number line, a parenthesis means that the starting number $(+3)$ is not included. Remember the number 3 is not part of the solution. If we plug in 3 into the inequality $x>3$ we would get $3>3$ which is not true.

Let’s try to graph another example. Let’s graph$x\le 5$. What does this mean? The key is to ask yourself what numbers you can replace x with that will give you a true statement? Will 8 work? No. $8\le 5$ Is not true. How about 1? Yes. 1 is a solution since $1\le 5$ is a true inequality. Is 1 the only solution? Wouldn’t 4.999 or 4 or 2.7 or 0 or $-164.9$ also work? They sure would. Plugging any of those numbers in for x will give a true statement. So we see that “$x\le 5$” represents “all real numbers less than or equal to 5. So we want to shade all the numbers to left of 5 on the number line.

Notice that the graph starts at 5 and shades all the numbers to the left. Notice also that it begins with a bracket. When graphing on the number line, a bracket means that the starting number $(+5)$ is included. The key is that the number 5 is part of the solution. If we plug in 5 into the inequality $x\le 5$ we would get $5\le 5$ which is true. So ”$x\le 5$” means “all real numbers less than or equal to 5”.

Inequalities really can be confusing when the variable is on the right hand side. For example look at $-1\ge x$. What does this mean? The key is still the same. What numbers can we replace x with that make this inequality true? Will $+7$ work? If we plug in $+7$ into the inequality $-1\ge x$ we get $-1\ge +7$. This is not true! $-1$ is less than $+7$. Let’s try another. Will $-4$ work? If we plug in $-4$ into the inequality $-1\ge x$ we get $-1\ge -4$. This is true! $-1$ is greater than $-4$. (A person that loses only $1 is richer than a person that loses $4.) What about $$-1.2\text{ or }-3\text{ or }-12.5\text{ or }-125\text{ or }-507.3?$$ Won’t they also make the inequality $-1\ge x$ true? Definitely. So what have we learned? Even though the inequality $-1\ge x$ uses a “greater than or equal to” symbol, in reality, “$-1\ge x$” means “all real numbers less than or equal to $-1$”.

Sometimes we like to represent all real numbers in between two numbers. Look at the inequality $-4<x<+2$. Notice the smallest number is on the left and the larger number is on the right. This is just like the number line. The key question again is what numbers can we replace x with that makes the inequality true? Will $+5$ work? No, $+5$ is not in between $-4$ and $+2$ on the number line. How about 0? Yes 0 is in between $-4$ and $+2$ on the number line. Can you name some more numbers in between? How about $-3.5$or $-2⁤\frac{5}{8}$ or $+1.997$? These would also make the inequality $-4<x<+2$ true. So the inequality “$-4<x<+2$” represents all real numbers in between $-4$ and $+2$ on the number line.

Notice again that it begins and ends with a parenthesis. When graphing on the number line, a parenthesis means that the starting number $(-4\text{ and }+2)$ are not included. If we plug in $-4\text{ or }+2$ into the inequality $-4<x<+2$ we would get statements that are not true.

How would the graph be different if the inequality was written as $-4\le x<+2$? Well notice that now $-4$ is now a solution, but $+2$ is still not. So we will need to use a bracket on $-4$ and a parenthesis on $+2$.

What do you think would happen if the inequality was $-4\le x\le +2$? If you said there would be brackets on both ends, you are absolutely right!

Sometimes you will see a not equal statement like $x\ne +5$ in algebra classes, but what does this mean? Think about what you can replace x with that would make it true. Notice you can plug in any number you want for x except $+5$. So $x\ne +5$ means “all real numbers except $+5$. What would the graph of $x\ne +5$ look like? Shade everything on the number line, but leave a parenthesis at $+5$.

Try the following inequality problems with your instructor.

Directions: Graph the following inequalities on the number line. Be careful to use either parenthesis and/or brackets correctly and shade the correct direction.

(Instructor examples continued.)

Directions: Explain each of the following graphs in words and write an inequality with “x” that represents it.

Sometimes it is necessary in algebra to solve an inequality for the variable. We are beginning to understand what inequalities like $x>7$ means, but what if it was $3x+1>7$? What does that mean? The key is to solve the inequality as if it was an equation. Imagine in your mind that $3x+1>7$ is just like $3x+1=7$. How would you solve that equation? Using the steps to solving equations, we would subtract 1 from both sides, and then divide by 3. This is exactly what you would do with the inequality as well.

$$\begin{array}{rcl}3x+1& >& 7\\ -1& & -1\\ 3x& >& 6\\ \frac{3x}{3}& >& \frac{6}{3}\\ 1x& >& 2\\ x& >& 2\end{array}$$

So we see that when we solve $3x+1>7$ for x we get $x>2$ which is all real numbers greater than 2.

A few important notes about solving inequalities

Though in most respects solving inequalities is the same as solving equations, there is 1 important difference. Look at the following example.

$$\begin{array}{l}-5<+7\text{ This is a true statement.}\\ -5++6<+7++6\text{ This is still true. When we add a positive number to both sides, }\\ \text{ the inequality remains true.}\\ -5-13<+7-13\text{ This is still true. When we subtract a number from both sides, }\\ \text{ the inequality remains true.}\\ -5+-10<+7+-10\text{ This is still true. When we add a negative number to both sides, }\\ \text{ the inequality remains true.}\\ \end{array}$$

So to summarize. We can add or subtract any number we want from both sides of the inequality and the inequality will remain true. Let’s look at another example.

$$\begin{array}{l}-15<+\text{10 This is a true statement.}\\ -15\times +4<+10\times +4\text{ This is still true. When we multiply both sides by a }\\ \text{ positive number, the inequality remains true.}\\ \frac{-15}{+5}<\frac{+10}{+5}\text{ This is still true. When we divide both sides by a }\\ \text{ positive number, the inequality remains true.}\\ -15\times -2<+10\times -\text{2 This is NOT true! When we multiply both sides by }-2\\ \text{ we get }+30<-\text{20. This is FALSE! So if you multiply }\\ \text{ both sides by a negative number, the inequality is false.}\\ \text{ Notice we can fix it. Just switch the inequality from }\\ <\text{ to > and we get true }\text{statement}+30>-20.\\ \frac{-15}{-5}<\frac{+10}{-5}\text{ This is also NOT true! When we divide both sides by }-5\\ \text{ we get }+3<-\text{2. This is FALSE! So if you divide }\\ \text{ both sides by a negative number, the inequality is false.}\\ \text{ Notice again we can fix it. Just switch the inequality from }\\ <\text{ to > and we get true }\text{statement}+3>-2.\end{array}$$

Rules for solving inequalities

• When solving inequalities, follow the same steps that we use to solve equations.
• If you add or subtract any number from both sides, the inequality remains true.
• If you multiply or divide both sides of the inequality by a positive number, the inequality remains true.
• If you multiply or divide both sides of the inequality by a negative number, the inequality is now false and must be switched to the opposite sign.

$$\begin{array}{l}<\text{ changes to }>\\ \le \text{ changes to }\ge \\ >\text{ changes to }<\\ \ge \text{ changes to }\le \end{array}$$

We have seen inequalities like $-3<w\le +11$ where the variable is in-between two numbers. We saw that this represents all real numbers between $-3$ and $+11$, including $+11$ but not including $-3$. What if there is an expression in the middle instead of a single variable? What does that mean? The key again is to solve the inequality and isolate the variable. Since there are three sides, we will be performing our solving steps to all three sides.

Remember, if you multiply or divide all three sides by a negative number, you must switch both signs. Look at the following example.

$$\begin{array}{l}-15<-2w+3\le +17\\ -15<-2w+3\le +17\text{ Subtract 3 from all three sides (or add }-3\text{ to all three sides)}\\ -3-3-3\\ -18<-2w+0\le +14\\ -18<-2w\le +14\\ \frac{-18}{-2}>\frac{-2w}{-2}\ge \frac{+14}{-2}\text{ Divide all three sides by }-2\text{ (Don't forget to switch the signs!!)}\\ +9>w\ge -7\end{array}$$

Solve the following inequalities with your instructor. Explain the answer in words.

Practice Problems Section 5G

Complete the following inequality problems. For #57–70, solve each inequality.

Directions: Identify which number is larger and which number is smaller or if the two numbers are equal. Then determine if the inequality symbol is used correctly or incorrectly?

1. $$+17>+12$$

   Check answer

   17 is larger than 12. The inequality is used correctly.

2. $$-5\ge +3$$

3. $$-7<-11$$

   Check answer

   −7 is larger than −11. The inequality is used incorrectly.

4. $$+17\le +17$$

5. $$+\frac{1}{4}\ne +0.25$$

   Check answer

   1/4 equals 0.25. The not-equal symbol is used incorrectly.

6. $$-25>0$$

7. $$+7⁤\frac{5}{6}\ge +7$$

   Check answer

   7 5/6 is larger than 7. The inequality is used correctly.

8. $$-84<-6$$

9. $$-21>-21$$

   Check answer

   The two values −21 are equal. The strict greater-than symbol is used incorrectly.

10. $$+0.274\ne +1.35$$

11. $$-19>+14$$

    Check answer

    14 is larger than −19. The inequality is used incorrectly.

12. $$+52\ge +52$$

13. $$-\frac{3}{4}<+\frac{1}{7}$$

    Check answer

    1/7 is larger than −3/4. The inequality is used correctly.

14. $$-13\le -13$$

15. $$+5.3\ne +7.6$$

    Check answer

    7.6 is larger than 5.3. The not-equal symbol is used correctly.

16. $$+5.3\ne +7.6$$

17. $$-9\le 0<+14$$

    Check answer

    −9 is less than 0, which is less than 14. Both comparison symbols are used correctly.

18. $$+3\le -10\le +21$$

19. $$-23<-23\le 0$$

    Check answer

    The first two values are equal at −23, and both are less than 0. The first strict less-than comparison is incorrect, so the chain is used incorrectly.

20. $$+3.9<+5.2\le +5.2$$

21. $$-7\ne +3$$

    Check answer

    3 is larger than −7. The not-equal symbol is used correctly.

Directions: Graph the following inequalities on the number line. Your number line should be labeled. Be careful to use either parenthesis and/or brackets correctly and shade the correct direction.

22. $$x>+6$$

23. $$x\le -12$$

    Check answer

    

24. $$-13\le x$$

25. $$x<+8$$

    Check answer

    

26. $$x\ge -11$$

27. $$+9\ge x$$

    Check answer

    

28. $$x<+2.5$$

29. $$x\ge \frac{1}{4}$$

    Check answer

    

30. $$-3⁤\frac{1}{2}\ge x$$

31. $$+14<x<+25$$

    Check answer

    

32. $$-6<x\le +7$$

33. $$0\le x\le +10$$

    Check answer

    

34. $$-13<x<-8$$

35. $$+1.5<x\le +5.5$$

    Check answer

    

36. $$\frac{1}{2}<x\le 2⁤\frac{1}{2}$$

37. $$-3.25\le x\le +1.75$$

    Check answer

    

38. $$x\ne +8$$

39. $$x\ne -1$$

    Check answer

    

40. $$x\ne +2.5$$

41. $$x\ne -3⁤\frac{1}{2}$$

    Check answer

    

42. $$x\ne -11$$

Directions: Explain each of the following graphs in words and write an inequality with “x” that represents it.

43. 

    Check answer

    $$x\le -2$$

44. 

45. 

    Check answer

    $$-5<x\le 3$$

46. 

47. 

    Check answer

    $$x\ge 5$$

48. 

49. 

    Check answer

    $$x\le -4$$

50. 

51. 

    Check answer

    $$-2\le x\le 2$$

In statistics, we often have to compare two decimals and determine if one is larger than another. One common example of this is P-value. We often want to know if the P-value is less than or greater than 0.05.

52. Is a P-value of 0.035 less or greater than 0.05? Use an inequality symbol to show your answer symbolically.

53. Is a P-value of 0.21 less or greater than 0.05? Use an inequality symbol to show your answer symbolically.

    Check answer

    $$0.21>0.05$$

54. Is a P-value of 0.0007 less or greater than 0.05? Use an inequality symbol to show your answer symbolically.

55. Is a P-value of $1.48\times {10}^{-6}$ less or greater than 0.05? Use an inequality symbol to show your answer symbolically.

    Check answer

    $$0.00000148<0.05$$

56. Is a P-value of $7.49\times {10}^{-2}$ less or greater than 0.05? Use an inequality symbol to show your answer symbolically.

57. $$x-3<5$$

    Check answer

    $$x<8$$

58. $$12\le x-5$$

59. $$m-7\le -27$$

    Check answer

    $$m\le -20$$

60. $$-4>b-1$$

61. $$-\frac{1}{4}n\le 2$$

    Check answer

    $$n\ge -8$$

62. $$6\le \frac{-3}{5}w$$

63. $$\frac{x}{2}<-111$$

    Check answer

    $$x<-222$$

64. $$20>-5c$$

65. $$-15c-30>15$$

    Check answer

    $$c<-3$$

66. $$4x-x+8\le 35$$

67. $$2x-3>2(x-5)$$

    Check answer

    All real numbers

68. $$7x+6\le 7(x-4)$$

69. $$-6<-3x<15$$

    Check answer

    $$-5<x<2$$

70. $$-3<2x-1<7$$

Adapted from Preparing for Algebra and Statistics, Third Edition, by Matt Teachout. License: CC BY 4.0. Copyright and sources.