Having finished lessons on basic equation solving, the next phase consists of using those same techniques to solve inequalities. I have not mentioned when to give quizzes or tests, since teachers will generally know when it is best to do so. But I break my rule here to say that now would be a good time to give a quiz on solving equations before moving on to inequalities.
Students know what inequalities are from previous grades, but it is good to review the basics in terms of expressing inequality relationships with variables. For example Jack weighs less than 50 pounds can be represented as j < 50, where j represents Jack’s weight.
Also it is important to know how to interpret sentences that pertain to “greater than or equal to”, and “less than or equal to”. These concepts are integrated in this initial lesson of solving inequalities.
Students find solving inequalities using addition or subtraction easy because it follows the same rules as equations which they have just studied. In my teaching, I have used this topic to serve not only as an extension of solving equations, but of solving word problems to provide students with more practice.
In this lesson we focus on inequalities with addition, subtraction and multiplication and division by positive numbers. The next lesson introduces inequalities when both sides are multiplied or divided by a negative number, which reverses the sign of the inequality.
Warm-Ups. Suggested type of warm-up problems follow:
1. Solve. 2n + 1/3 = 7/9 Answer: 2n = 4/9; n = 4/9 ∙ 1/2 =2/9
2. Write an equation and solve. Janet has three times as much money as Peter. Together they have $40. How much money do each of them have?
Answer: If x = amt of Peter’s money, 3x is amt of Janet’s money. 3x + x = 40; 4x=40; x = $10, 3x = $30.
3. Distribute. 2/3(3 + 9m) Answer: 2 + 6m
4. Write an equation and solve. Nine more than two times a number is 25. What is the number? Answer: 9+2x = 25; 2x = 16; x = 8
5. Translate to algebra. Extra credit if you solve! Nine less than two times a number is less than or equal to 57. Answer: 2y-9 ≤ 57; 2y≤64; y≤32
In going through the warm-ups, Problem 5 will be the only one students have not had experience with. Hints to students may include, “How would you translate it if it said ‘equals 57’? Can you substitute a different symbol?”
Basic Review. It will not take much review for students to recall the symbols < and >. Also, when going over Problem 5 of the warm-ups students will then recall the symbols ≤ and ≥. The critical aspect of the review is to now abridge that knowledge with translation of English into algebra. For example, I ask students “If x represents the amount of money John has, how would I write this in symbols: ‘John has less than $40’ ?” Students usually have no problem telling me it is x < 40.
They typically have more difficulty with the concepts of “less than or equal to” and “greater than or equal to”. I have used the example of minimum height to go on amusement park rides. The phrase “You must be at least 48 inches to go on this ride” would translate to x ≥ 48. I scaffold this by asking “Does ‘at least’ mean you have to be greater or less than 48 inches? What if someone is exactly 48 inches—can they go on the ride?”
Similarly for these type of phrases:
Jerry can spend no more than $20 on candy. Answer: j ≤ 20
Suggested hints: If it’s “no more than $20” can it be $21? Can it be $19? Can it be $20?
To ride on the Wild Jaguar you must be at most 72 inches in height. Answer: m ≤ 72
Jeanine has lost at least 17 pounds. Answer: x ≥ 17
You must have a score no less than 95% to qualify. g ≥ 95
The speed on the turnpike can be no more than 70 mph. s ≤ 70
Solving Inequalities by Addition or Subtraction. The principles of the addition and subtraction properties of equality also apply to inequalities. To illustrate, I will write 15 < 22. If I add 3 to each side I then have 18 < 25. If I subtract 7 from each side I have 8 < 14. The inequality is preserved in either case.
Formally, it is stated as follows:
Addition Property of Inequality: If a < b, then a+c < b+c
Subtraction Property of Inequality: If a < b, then a-c < b-c
A few examples will convince them that these problems are basically no different than what they have been doing with one-step equations, except for the inequality signs.
3 + m < 7; Answer: m < 4
y- 4/5 ≤ 5/6; Answer: y ≤ 1/30
Graphing Inequalities. A graph of an inequality indicates the portion of a number line that represents all possible solutions to that inequality. In this way, students have a visual representation that inequalities have an infinite amount of solutions. For example, the statement y < 6 is represented by a number line in which all numbers to the left of 6 are shaded, and the number 6 itself is indicated with a hollow dot: o
(Image was found; ignore that billboard like announcement above!)
The statement y ≤ 6 is similar except that the number 6 is indicated with a solid dot to show that it is included as a solution.
Solving Inequalities with Multiplication and Division. Multiplying or dividing each side of an inequality by a positive number will preserve the inequality. Again, students will find that solving either one- or two-step inequalities is no different than solving equations.
An inequality such as 3 < 15 is preserved when both sides are multiplied by 10:
30 < 150
Similarly, the inequality is preserved when both sides are divided by 3:
1 < 5
Formally, the rule is stated as follows:
If a < b and c > 0 ( is positive), then a∙c < b∙c and a/c < b/c
Examples: 2x < 40; Answer: x < 40
3x - 5 ≤ 22 Answer: 3x ≤ 27; x ≤ 9
x/5 ≥ 3 x ≥ 15
m + 3m > 56 Answer: 4m > 56; m > 14
Upping the ante to word problems:
Translate to algebra and solve. Three less than 4 times a number is less than 325. Answer: 4h -3 < 325; 4h < 328; h<82
Five more than a number added to the number is greater than eighteen. Answer: 5 +x + x > 19; 5 + 2x < 19; 2x < 14; x < 7
Four times a number added to the number is no greater than twenty five. Answer: 4m + m ≤ 25; 5m ≤ 25; m ≤ 5
Now we up the ante a little further with more substantial (i.e., wordy) word problems.
Asher has $10 to spend at a bowling party. If each game costs $2, up to how many games can he bowl? Write an inequality and solve. Answer: Let x = the maximum number of games. 2x ≤ 10; x ≤ 5. He can therefore bowl up to five games.
A similar problem but slightly more difficult problem should follow.
Allen works for a lawn service. It takes him 1 ½ hours to mow a lawn. Write and solve an inequality to find the number of lawns he can mow if he works at least 15 hours.
This will take some prompting. “What is it we’re trying to find?” The number of lawns he can mow. “If he mowed 2 lawns, how do we find out how long it takes?” They will see it is 2 times 1 ½ . If 3 hours, it’s 3 times 1 ½ . “Since we don’t know how long it takes, what do we multiply by 1 ½ ?” Now we have 1 ½ x, which is easier to write as (3/2)x. “If it were to equal exactly 15 hours, what would we write?”
It would be (3/2)x = 15. But it’s an inequality. He must work at least 14 hours. “What symbol do we use?” It should be (3/2)x ≥ 15. Solving this, x ≥ 10 Therefore, Allen would mow at least 10 lawns; 10 if he worked exactly 15 hours, and more if he worked more than 15 hours.
Homework. The homework, which is to be started in class continues these types of problems, which should include some graphing, and most definitely some word problems—both the numerical type, and more substantive ones. The main confusion students will have is with phrases such as “at least”, “at most”, “no more than”, and “no less than”.