Name____________________________ Period______ Date_________
Linear Functions – Activity 3.5 – Linear Inequalities
How Long Can You Live?: Life expectancy in the United States is steadily increasing, and the number of
Americans aged 100 or older will exceed 850,000 by the middle of this century. Medical advancements
have been a primary reason for Americans living longer. Another factor has been the increased
awareness of maintaining a healthy lifestyle.
The life expectancies at birth for mean and women born after 1980 in the United States can be modelled
by the following functions.
𝑊(𝑥) = 0.115𝑥 + 77.42
𝑀(𝑥) = 0.212𝑥 + 69.80
Where 𝑊(𝑥) represents the life expectancy for women, 𝑀(𝑥) represents the life expectancy for men,
and 𝑥 represents the number of years since 1980 that the person was born. That is, 𝑥 = 0, corresponds
to the year 1980; 𝑥 = 5 corresponds to 1985 and so forth.
1.
a. Complete the following table.
𝑥, years since 1980
1980
0
1985
5
Year
1990
10
1995
15
2000
20
2005
25
𝑊(𝑥)
𝑀(𝑥)
a. For people born between 1980 and 2005, do men or women have the greater life
expectancy?
b. Is the life expectancy of men or women increasing more rapidly? Explain using slope.
You would like to determine in what birth years the life expectancy of men is greater than that of
women. The phrase “greater than” indicates a mathematical relationship called an inequality.
Symbolically, the relationship can be represented by
𝑀(𝑥)
Life expectancy
for men
>
is greater than
𝑊(𝑥)
Life expectancy
for women
1
Other commonly used phrases that indicate inequalities are given in the following example.
.
2. Substitute the appropriate expressions for 𝑀(𝑥) and 𝑊(𝑥) to obtain an inequality involving 𝑥 that
can be used to determine the birth years for which the life expectancy of men is greater than that of
women.
Solving Inequalities in One Variable Numerically and
Graphically
Definition - Solving an inequality in one variable is the process of determining the values of the
variable that make the inequality a true statement. These values are called the solutions of the
inequality.
3. Solve the inequality in Problem 2 numerically. That is continue to construct a table of values (see
problem 1) until you determine the values of the years 𝑥 (inputs) for which
0.212𝑥 + 69.80 < 0.115𝑥 + 77.42. Use the table feature of your graphing calculator.
i.
ii.
iii.
iv.
v.
Enter /I 2 and you will see a blank graph
Enter after the = sign f1(x)=0.212𝑥 + 69.80
/G
Then after the = sign, f2(x)=0.115𝑥 + 77.42
b, 7, 1 to see the table
2
Scroll through the table to find the year their life expectancy is close to equal. The year the life
expectancy for men exceeds women is 1980+______ = _________ with a life expectancy of
approximately _________ years.
Therefore if the trends given by the equations fro 𝑀(𝑥) and 𝑊(𝑥) continue, the approximate solution
to the inequality 𝑀(𝑥) > 𝑊(𝑥) is 𝑥 > 79. That is, according to the models, after the year 2059, men
will live longer than women.
4. Solve the inequality 0.212𝑥 + 69.80 < 0.115𝑥 + 77.42 graphically. You may view the graph below
or do this on your own. Hit /e to go back to the empty graph. Enter b41 and enter an
Xmax and Ymax based on your values from above to display the graph.
i.
You will see the graph of the 2 lines.
ii.
Enter b, 6, 4
iii.
Pick point to the left of the intersection and hit enter
iv.
Pick a point to the right of the intersection and you will see the intersection point
a. What is the point of intersection (
situation?
,
)? What does the point represent in this
5. To solve the inequality 𝑀(𝑥) > 𝑊(𝑥) graphically, you need to determine the value of 𝑥 for which
the graph of 𝑀(𝑥) = 0.212𝑥 + 69.80 is above the graph of 𝑊(𝑥) > 0.115𝑥 + 77.42
a. Use the graph below to determine when 𝑀(𝑥) > 𝑊(𝑥). How does your solution compare
to the solution using the table?
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Solving Inequalities in One Variable Algebraically
The process of solving an inequality in one variable algebraically is very similar to solving an equation in
one variable algebraically. Your goal is to isolate the variable on one side of the inequality symbol. You
isolate the variable in an equation by performing the same operations to both sides of the equation so
as not to upset the balance. You isolate the variable in an inequality by performing the same operations
to both sides so as not to upset the imbalance.
6. a. Write the statement “15 is greater than 6” as an inequality.
b. Add 5 to each side of 15 > 6. Is the resulting inequality a true statement? (That is, is the left
side still greater than the right side?)
c. Subtract 10 from each side of 15 > 6. Is the resulting inequality a true statement? (That is, is
the left side still greater than the right side?)
d. Multiple each side of 15 > 6 by 4. Is the resulting inequality true?
e. Multiply each side of 15 > 6 by −2. Is the left side still greater than the right side?
f.
Reverse the direction of the inequality symbol in part e. Is the new inequality a true statement?
Problem 6 demonstrates two very important properties of inequalities.
Property 1 If 𝑎 < 𝑏 represents a true inequality, then if
i.
ii.
The same quantity is added to or subtracted from both sides, or
Both sides are multiplied or divided by the same positive number, then the resulting
inequality remains a true statement and the direction of the inequality symbol remains the
same.
For example, because −4 < 10, then
1. −4 + 5 < 10 + 5 or 1 < 15 is true.
−4 − 3 < 10 − 3 or − 7 < 7 is true.
2. −4(6) < 10(6)or − 24 < 60 is true.
4 10
− <
or − 2 < 5 is true.
2
2
4
Property 2 If 𝑎 < 𝑏 represents a true inequality, then if both sides are multiplied or divided by the
same negative number, then the inequality symbol in the resulting inequality statement must be
reversed (< to > or > to <) in order for the resulting statement to be true.
For example, because −4 < 10, then −4(−5) > 10(−5) or 20 > −50.
Because −4 < 10, then
−4
−2
>
10
or 2
−2
> −5
Properties 1 and 2 will be true if 𝑎 < 𝑏 is replaced by 𝑎 ≤ 𝑏, 𝑎 > 𝑏, or 𝑎 ≥ 𝑏.
The following example demonstrates how properties of inequalities can be used to solve an inequality
algebraically.
Solve 3(𝑥 − 4) > 5(𝑥 − 2) − 8
SOLUTION
3(𝑥 − 4) > 5(𝑥 − 2) − 8
3𝑥 − 12 > 5𝑥 − 10 − 8
3𝑥 − 12 > 5𝑥 − 18
−5𝑥
− 5𝑥
−2𝑥 − 12 > −18
+12
+ 12
−2𝑥 > −6
−
2𝑥
2
6
> −2
apply the distributive property
combine like terms on the right side
subtract 5𝑥 from both sides; the direction of the inequality symbol
remains the same
add 12 to both sides; the direction of the inequality symbol
does not change
divide both sides by −2; the direction is reversed.
𝑥<3
Therefore, from Example 2, any number less than 3 is a solution to the inequality
3(𝑥 − 4) > 5(𝑥 − 2) − 8. The solution set can be represented on a number line by shading all points to
the left of 3:
The open circle at 3 indicates that 3 is not a solution. A closed circle indicates that the number beneath
the closed circle is a solution. The arrow show that the solutions extend indefinitely to the left.
7. Solve the inequality 0.212𝑥 + 69.80 < 0.115𝑥 + 77.42 algebraically to determine the birth years
in which men will be expected to live longer than women. How does your solution compare to the
solution determined numerically and graphically in problems 3 and 4c?
0.212𝑥 + 69.80 < 0.115𝑥 + 77.42
5
Compound Inequality
You have joined a health-and-fitness club. Your aerobics instructor recommends that to achieve the
most cardiovascular benefit from your workout, you should maintain your pulse rate between a lower
and upper range of values. These values depend on your age.
8. If the variable 𝑎 represents your age, then the lower and upper values for your pulse rate and
determined by the following.
lower value: 0.72(220 − 𝑎)
upper value: 0.87(220 − 𝑎)
a. Determine your lower value.
b. Determine your upper value.
For the most cardiovascular benefit, a 20 year-old’s pulse rate should be between 144 and 174. The
phrase between “144 and 174” means the pulse rate should be greater than 144 and less than 174.
Symbolically, this combination or compound inequality is written as
144 < pulse rate 𝑎𝑛𝑑 pulse rate < 174
This pulse rate is written more compactly as 144 < pulse rate < 174
The numbers that satisfy this compound inequality can be represented on a number line as
Other commonly used phrases that indicate compound inequalities involving the word and are given in
the following example.
6
9. Recall that the life expectancy for men is given by the expression 0.212𝑥 + 69.80, where 𝑥 represents
the number of years since 1980. Use this expression to write a compound inequality that can be used
to determine in what birth years men will be expected to live into their 80’s.
The following example demonstrates how to solve a compound linear inequality algebraically and
graphically.
Solve −4 < 3𝑥 + 5 < −11 using an algebraic approach.
SOLUTION
Note that the compound inequality has three parts: left: −4, middle 3𝑥 + 5, and right: 11. To solve this
inequality, isolate the variable in the middle part.
−4 < 3𝑥 + 5 < −11
−5
− 5 − 5 Subtract 5 from each part.
−9 < 3𝑥 ≤ 6
9
3𝑥
3
3
− <
≤
6
3
Divide each part by 3
−3 < 𝑥 < 2
The solution can be represented on a number line as follows:
10. Solve the compound inequality 80 ≤ 0.212𝑥 + 69.8 < 90 from problem 9 to determine in what
birth years men will be expected to live into their 80’s.
Don’t forget to add the solution to 1980.
7
Interval Notation
Interval notation is an alternate method to represent a set of real numbers, described by an inequality.
The closed interval [−3,4] represents all real numbers 𝑥 for which −3 ≤ 𝑥 ≤ 4. The square brackets
[ ] indicate that the endpoints of the interval are included. The open interval (−3,4) represents all
real numbers 𝑥 for which −3 < 𝑥 < 4. Note that the ( ) indicate that the endpoints of the interval are
not included. The interval (−3,4] is said to be half-open or half-closed. The interval is open at −3
(endpoint not included) and closed at 4 (endpoint included).
Suppose you want to represent the set of real numbers 𝑥 for which 𝑥 is greater than 3. The symbol +∞
(positive infinity) is used to indicate unboundedness in the positive direction. Therefore, the interval
(3, +∞) represents all real numbers 𝑥 for which 𝑥 > 3. Note that +∞ is always open.
The symbol −∞ (negative infinity) is used to indicate unboundedness in the negative direction.
Therefore, the interval (−∞, 5] represents all real numbers 𝑥 for which 𝑥 ≤ 5.
11. In parts a-d, ex\press each inequality in interval notation.
𝐚. −5 ≤ 𝑥 ≤ 10
𝐛. 4 ≤ 𝑥 < 8.5
𝐜. 𝑥 > −2
𝐝. 𝑥 ≤ 3.75
In parts e-h, express each of the following using inequalities.
𝐞. (−6,4]
𝐟. (−∞, 1.5]
𝐠. (−2, 2)
𝐡. (−3, +∞)
Summary - Inequalities
1. The solution set of an inequality is the set of all values of the variable that satisfy the inequality.
2. The direction of an inequality is not changed when
i.
The same quantity is added to or subtracted from both sides of the inequality. Stated symbolically,
If 𝑎 < 𝑏 then 𝑎 + 𝑐 < 𝑏 + 𝑐 and 𝑎 − 𝑐 < 𝑏 − 𝑐.
ii.
The same quantity is multiplied to or divided on both sides of the inequality. Stated symbolically,
𝑎
𝑏
If 𝑎 < 𝑏 then 𝑎𝑐 < 𝑏𝑐 and 𝑐 < 𝑐 whenever 𝑐 > 0.
3.
The direction of an inequality is reversed if both sides of an inequality are multiplied by or divided by the
same negative number. These properties can be written symbolically as
i.
If 𝑎 < 𝑏, then 𝑎𝑐 > 𝑏𝑐 where 𝑐 < 0
ii.
𝑎
𝑏
𝑐
𝑐
If 𝑎 < 𝑏, then > where 𝑐 < 0
The two properties of inequalities above (items 2 and 3) will still be true if 𝑎 < 𝑏 is replaced by 𝑎 ≤ 𝑏, 𝑎 > 𝑏,
or 𝑎 ≥ 𝑏
4. Inequalities
suchofasthe
𝑓(𝑥)
< 𝑔(𝑥) equations.
can be solved using three different methods
1. Solve each
following
i.
A numerical approach, in which a table of input/output pairs is used to determine values of 𝑥 for
which 𝑓(𝑥) < 𝑔{𝑥}:
ii.
A graphical approach, in which values of 𝑥 are located so that the graph of 𝑓 is below the graph of 𝑔.
iii.
An algebraic approach, in which the properties of inequalities are used to isolate the variable.
Practice
Similar statements can be made for solving inequalities of the form
𝑓(𝑥) ≤ 𝑔(𝑥), 𝑓(𝑥) > 𝑔(𝑥), and 𝑓(𝑥) ≥ 𝑔(𝑥).
8
Practice
In exercise 1-6, translate the given statement into an algebraic inequality or compound inequality.
1. To avoid an additional charge, the sum of the length, 𝑙, width 𝑤, and depth 𝑑, of a piece of luggage
to be checked on a commercial airline can be at most 61 inches.
2. A PG-13 movie rating mean that your age, 𝑎, must be at least 13 years for you to view the movie.
3. The cost 𝐶(𝐴), of renting a car from company 𝐴 is less expensive that the cost, 𝐶(𝐵), of renting
from company 𝐵.
4. The label on a bottle of file developer states that the temperature, t, of the contents must be kept
between 68° and 77° Fahrenheit.
5. You are in a certain tax bracket if your taxable income, 𝑖, is over $24,650, but not pver $59,750.
6. The range of temperature, t, on the surface of Mars is from 28° to 140° C.
Solve Exercises 7-14 graphically and algebraically, then check your solution set to make sure the arrow is
pointing in the correct direction.
7. 3𝑥 > −6
8. 3 − 2𝑥 ≤ 5
Check:
Check:
9. 𝑥 + 2 > 3𝑥 − 8
10. 5𝑥 − 1 ≤ 2𝑥 + 11
Check:
Check:
9
11. 8 − 𝑥 ≥ 5(8 − 𝑥)
12. 5 − 𝑥 < 2(𝑥 − 3) + 5
Check:
Check:
13.
𝑥
2
+ 1 ≤ 3𝑥 + 2
Check:
14. 0. 5𝑥 + 3 ≥ 2𝑥 − 1.5
Check:
Solve Exercises 15-16 algebraically. Check your solution set.
𝑥
15. 1 < 3𝑥 − 2 < 4
16. −2 < 3 + 1 < 5
Check:
Check:
10
17. The consumption of cigarettes in the United States is declining. If 𝑡 represents the number of years
since 1990. Then the consumption, 𝐶, is modeled by
𝐶 = −9.90𝑡 + 529.54
where 𝐶 represents the number of billions of cigarettes smoked per year.
a. Write an inequality that can be used to determine the first year in which cigarette consumption
is less than 200 billion cigarettes per year.
b. Solve the inequality in part a using an algebraic approach.
18. In Activity 7, Moving Out, you contacted two local rental companies and obtained the following
information for the 1-day cost of renting a truck.
Company 1: $19.99 per day plus $0.79 per mile
Company 2: $29.99 per day plus $0.59 per mile
Let 𝑛 represent the total number of miles driven in one day.
a. Write an expression to determine the total cost, C of renting a truck for 1 day from company 1.
b. Write an expression to determine the total cost, C of renting a truck for 1 day from company 2.
c. Use the expressions in parts a and b to write an inequality that can be used to determine for
what number of miles it is less expensive to rent the car from company 2.
d. Solve the inequality.
11
19. The sign on the elevator in a seven story building on campus states that the maximum weight it can
carry is 1200 pounds. As part of your work-study program, you need to move a large shipment of
books to the sixth floor. Each box weighs 60 pounds.
a. Let 𝑛 represent the number of boxes placed in the elevator. If you weigh 150 pounds, write
an expression that represents the total weight in the elevator. Assume that only you and the
boxes are in the elevator.
b. Using the expression in part a, write an inequality that can be used to determine the
maximum number of boxes that you can place in the elevator at one time.
c. Solve the inequality.
20. The following equation is used in meteorology to determine the temperature humidity index 𝑇: 𝑇 =
2
(𝑤
5
+ 80) + 15.
where 𝑤 represents the wet-bulb thermometer reading. For what values of 𝑤 would 𝑇 range from
70 to 75?
21. The temperature readings in the United States have ranged from a record low of −79.8℉ ( Alaska,
January 23, 1971) to a record high of 134℉ (California, July 10, 1913)
a. If 𝐹 represents the Fahrenheit temperature, write a compound inequality that
represents the interval of temperatures (in℉) in the United States.
b. Recall that Fahrenheit and Celsius temperatures are related by the formula
𝐹 = 1.8𝐶 + 32
12
Rewrite the compound inequality in part a to determine the temperature range in
degrees Celsius.
c. Solve the compound inequality.
22. You are enrolled in a wellness course at your college. You achieved grades of 70, 86, 81, and 83 on
the first four exams. The final exam counts the same as an exam given during the semester.
a. If 𝑥 represents the grade on the final exam, write an expression that represents your
course average (arithmetic mean).
b. If the average is greater than or equal to 80 and less than 90, you will earn a B in the
course. Using the expression from part a for your course average, write a compound
that must be satisfied to earn a B.
c. Solve the inequality.
13