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34 Lesson 1.8 ~ Linear Inequalities in One Variable
Nathan has more than $10 in his wallet. Jackie has run at most
200 miles this year. Each of these statements can be written using
an inequality. Inequalities are mathematical statements which use
>, $10 j ≤ 200Inequalities have multiple answers that can make
the statement true. In Nathan's example, he might have $20 or $100.
All that is known for certain is that he has more than $10 in his
wallet. In Jackie's example, she might have run 200 miles this year
or 5 miles. There are an infinite number of possibilities that make
each statement true.
write an inequality for each statement.a. Carla's weight (w) is
greater than 100 pounds.b. Vicky has at most $500 in her savings
account. Let m represent the amount of money in Vicky's account.c.
quinton's age is greater than 40 years old. Let a represent
quinton's age.
a. The key words are “greater than”. Use the symbol >. w >
100
b. The key words are “at most”. This means she has less than or
equal to $500. Use the ≤ symbol. m ≤ 500 c. The key words are
“greater than”. Use the > symbol. a > 40
linEar inEqualiTiEs in onE variablE
Lesson 1.8
ExamplE 1
solutions
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Lesson 1.8 ~ Linear Inequalities in One Variable 35
It is not possible to list all of the solutions to an
inequality. In example 1c, Quinton could be 41, 42 or even 43.5
years old. All the answers can be shown on a number line.
a > 40
39 40 41 42 43 44 45 46
Forty is not included in the solution because Quinton's age is
greater than 40, not equal to 40. This is shown on the number line
with an "open circle" at 40. All of the numbers to the right of 40
are greater than 40 so they are included in the solution. This is
shown with a line and an arrow pointing to the rest of the
values.
When using the > or < inequality symbols, an "open circle"
is used on the number line because the solution does not include
the given number. When using the ≥ or ≤ inequality symbols, a
"closed (or filled in) circle" is used because the solution
contains the given number. Determining which direction the arrow
should point is based on the relationship between the variable and
the solution. The arrow points towards the set of numbers that make
the statement true.
Inequalities are solved using properties similar to those you
used to solve equations. Use inverse operations to isolate the
variable so the solution can be graphed on a number line.
solve the inequality and graph its solution on a number line. x
_ 4 + 2 ≥ 3 Subtract 2 from both sides of the inequality. x _ 4 + 2
≥ 3 −2 −2 Multiply both sides of the inequality by 4. 4 ∙ x _ 4 ≥ 1
∙ 4 x ≥ 4 Graph the solution on a number line. Use a closed
circle.
−1 0 1 2 3 4 5 6
ExamplE 2
solution
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36 Lesson 1.8 ~ Linear Inequalities in One Variable
solve the inequality and graph its solution on a number line. 6x
+ 3 < 2x − 5
Subtract 2x from each side of the inequality. 6x + 3 < 2x − 5
−2x −2x 4x + 3 < −5Subtract 3 from each side. −3 −3 4
4x < 8
4−
Divide both sides by 4. x < −2
Graph the solution on a number line. Use an open circle.
−5 −4 −3 −2 −1 0 1 2
One special rule applies to solving inequalities. Whenever you
multiply or divide by a negative number on both sides of the
equation, you must flip the inequality symbol. For example, less
than () if you multiply or divide by a negative number.
solve the inequality −4x + 7 ≤ 19. Subtract 7 from each side of
the inequality. −4x + 7 ≤ 19 −7 −7Divide both sides by −4. −4x ____
−4 ≤
12 ___ −4 Since both sides were divided by a negative,flip the
inequality symbol. x ≥ −3
ExErcisEs
write an inequality for each graph shown. use x as the variable.
1.
−4 −3 −2 −1 0 1 2 3 2.
−3 −2 −1 0 1 2 3 4
3. −3 −2 −1 0 1 2 3 4
4. −3 −2 −1 0 1 2 3 4
ExamplE 3
solution
ExamplE 4
solution
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Lesson 1.8 ~ Linear Inequalities in One Variable 37
solve each inequality. Graph the solution on a number line. 5.
4x − 1 ≥ 15 6. 10 < 6 + 2x 7. 3x + 10 < −2
8. 5x − 7 > 8 9. x __ 2 − 1 ≥ −1 10. −3x − 4 < 5 11. 3(x +
1) ≥ 9 12. 3 < 1 + x ___ −4 13. 5x < 2x − 21
14. −2 + 4x ≥ 3 − 6x 15. 2(x + 3) ≥ 5x + 12 16. 1 _ 2 x + 2 <
x
17. A forklift has a maximum carrying capacity of 960 pounds.
Each cargo box weighs 60 pounds. a. Write and solve an inequality
that represents the maximum number of cargo boxes the forklift can
hold. b. A 120-pound carrying case is used to hold the cargo boxes.
What is the maximum number of cargo boxes the forklift can carry
when the carrying case is used? Show that your answer is correct by
showing that one more than your answer would exceed the forklift's
capacity.
18. Olivia has $700 in her bank account at the beginning of the
summer. She wants to have at least $150 in her account at the end
of the summer. Each week she withdraws $40 for food and
entertainment. a. Write an inequality for this situation. Let x
represent the number of weeks she withdraws money from her account.
b. What is the maximum number of weeks that Olivia can withdraw
money from her account? Explain how you know your answer is
correct. c. How much money will be left in her account after the
last full withdrawal?
19. Ivan was at the beach. He wanted to spend $12 or less on a
beach bike rental. The company he chose to rent from charged an
initial fee of $5 and an additional $0.45 per mile he rode. a.
Write an inequality for this situation. Let x represent the number
of miles ridden. b. How many miles can Ivan ride without going over
his spending limit? Write your answer as a whole number.
20. Mike went to the arcade. He spent less than or equal to $30;
but spent more than $25. Create a number line that shows all the
possible amounts that Mike may have spent at the arcade. Use words
and/or numbers to show how you determined your answer.
rEviEw
solve each equation. describe the number of solutions (one, none
or infinitely many). 21. 4x − 3 = x + 18 22. 4(x − 1) = 4x − 8 23.
5x − 3x + 2 = 2x + 2
24. 1 _ 2 (x + 4) = 1 _ 2 x + 2 25. 6(x − 3) = x − 3 26. 9 + 6x
= 6x + 2 + 4
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38 Lesson 1.8 ~ Linear Inequalities in One Variable
tic-tAc-toe ~ com poun d inequA l iti e s
Compound inequalities contain two inequality symbols. An example
of a compound inequality is 4 < x < 9. This can be read “x is
greater than four and less than 9.” A solution to a
compound inequality is a value that makes the statement true. In
this case, any number between 4 and 9 makes the statement true. For
example, 5.5 is a solution, 4 < 5.5 < 9, because it is
greater than 4 but less than 9.
To graph a compound inequality, place open or closed circles
(depending on the inequality) on each end value and connect the
circles with a line in between.
0 5 10When solving compound inequalities, you must isolate the
variable in the middle part of the inequality. In order to maintain
the balance of the inequality, you must perform operations on ALL
THREE parts of the inequality (left, middle and right)
example: Solve 2 ≤ x − 1 < 11. Graph the solution. 2 ≤ x − 1
< 11 Add 1 to all three parts of the inequality. +1 +1 +1 3 ≤ x
< 12
0 5 10
solve each inequality below. Graph the solution on a number
line. 1. 2 ≤ x + 2 ≤ 6 2. 6 < 2x < 16 3. −5 ≤ x − 2 <
−3
4. 5 < 3x + 5 ≤ 17 5. 1 < x _ 3 − 1 < 2 6. 1
_ 2 ≤ 2x − 3 ≤ 1
tic-tAc-toe ~ one doe s not Be longCreate ten game cards. Each
game card needs three equations on it that require two or more
steps to solve. Two of the equations on a card should represent
equations that have the same number of solutions (one, none or
infinitely many). The other equation should have a different number
of solutions. Participants try to locate the equation that does not
fit. The cards can be used as a game or as a full class activity
like a warm-up. Change the placement of the equations that do not
belong so they are not always in the same spot on the cards.