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Notes by Dr. David Archerteacher of Calculus at Andress High
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ALGEBRA PROJECT
UNIT 6
SOLVING LINEAR INEQUALITIES
SOLVING LINEAR INEQUALITIES
Lesson 1Solving Inequalities by Addition and Subtraction
Lesson 2 Solving Inequalities by Multiplication and Division
Lesson 3 Solving Multi-Step Inequalities
Lesson 4 Solving Compound Inequalities
Lesson 5Solving Open Sentences Involving Absolute Value
Lesson 6 Graphing Inequalities in Two Variables
SOLVING INEQUALITIES by ADDITION and SUBTRACTION
Example 1 Solve by Adding
Example 2 Graph the Solution
Example 3 Solve by Subtracting
Example 4 Variables on Both Sides
Example 5 Write and Solve an Inequality
Example 6 Write an Inequality to Solve a Problem
Answer: The solution is the set {all numbers greater than 77}.
Solve Then check your solution.
Original inequality
Add 12 to each side.This means all numbers greater than 77.
Check Substitute 77, a number less than 77, and anumber greater than 77.
Solve Then check your solution.
Answer: or {all numbers less than 14}
Solve Then graph it on a number line.
Original inequality
Add 9 to each side.Simplify.
Answer: Since is the same as y 21, the solution set is
The dot at 21 shows that 21 is included in the inequality.
The heavy arrow pointing to the left shows that the inequality includes all the numbers less than 21.
Solve Then graph it on a number line.
Answer:
Solve Then graph the solution.
Original inequality
Subtract 23 from each side.Simplify.
Answer: The solution set is
Solve Then graph the solution.
Answer:
Then graph the solution.
Original inequality
Subtract 12n from each side.Simplify.
Answer: Since is the same as the solution set is
Then graph the solution.
Answer:
Write an inequality for the sentence below. Then solve the inequality.
Seven times a number is greater than 6 times that number minus two.
Seven timesa number
is greaterthan
six timesthat number minus two.
7n 6n 2> –
Simplify.Subtract 6n from each side.
Original inequality
Answer: The solution set is
Write an inequality for the sentence below. Then solve the inequality.
Three times a number is less than two times that number plus 5.
Answer:
Entertainment Alicia wants to buy season passes to two theme parks. If one season pass cost $54.99, and Alicia has $100 to spend on passes, the second season pass must cost no more than what amount?
Words The total cost of the two passes must be less than or equal to $100.
Variable Let the cost of the second pass.
Inequality 100The total cost
is less thanor equal to $100.
Solve the inequality.
Answer: The second pass must cost no more than $45.01.
Original inequality
Subtract 54.99 from each side.
Simplify.
Michael scored 30 points in the four rounds of the free throw contest. Randy scored 11 points in the first round, 6 points in the second round, and 8 in the third round. How many points must he score in the final round to surpass Michael’s score?
Answer: 6 points
SOLVING INEQUALITIES by MULTIPLICATION and DIVISION
Example 1 Multiply by a Positive Number
Example 2 Multiply by a Negative Number
Example 3 Write and Solve an Inequality
Example 4 Divide by a Positive Number
Example 5 Divide by a Negative Number
Example 6 The Word “not”
Then check your solution.
Original inequality
Multiply each side by 12. Since we multiplied by a positive number, the inequality symbol stays the same.
Simplify.
Check To check this solution, substitute 36, a number less that 36 and a number greater than 36 into the inequality.
Answer: The solution set is
Then check your solution.
Answer:
Original inequality
Simplify.
Multiply each side by and change
Answer: The solution set is
Answer:
Write an inequality for the sentence below. Then solve the inequality.
Four-fifths of a number is at most twenty.
Four-fifths of is at most twenty.a number
r 20
Answer: The solution set is .
Original inequality
Simplify.
Multiple each side by and do not
change the inequality’s direction.
Write an inequality for the sentence below. Then solve the inequality.Two-thirds of a number is less than 12.
Answer:
Original inequality
Divide each side by 12 and do not change the direction of the inequality sign.
Simplify.
Check
Answer: The solution set is
Answer:
using two methods.
Method 1 Divide.
Original inequality
Divide each side by –8 and change < to >.
Simplify.
Answer: The solution set is
Method 2 Multiply by the multiplicative inverse.
Original inequality
Multiply each side byand change < to >.
Simplify.
using two methods.
Answer:
Multiple-Choice Test Item
Which inequality does not have the solution
A B C D
Read the Test ItemYou want to find the inequality that does not have the solution set
Solve the Test Item
Consider each possible choice.
A.
D. C.
B.
Answer: B
Multiple-Choice Test Item
Which inequality does not have the solution ?
A B C D
Answer: C
SOLVING MULT-STEP INEQUALITIES
Example 1 Solve a Real-World Problem
Example 2 Inequality Involving a Negative Coefficient
Example 3 Write and Solve an Inequality
Example 4 Distributive Property
Example 5 Empty Set
Science The inequality F > 212 represents the
temperatures in degrees Fahrenheit for which water is
a gas (steam). Similarly, the inequality
represents the temperatures in degrees Celsius for
which water is a gas. Find the temperature in degrees
Celsius for which water is a gas.
Answer: Water will be a gas for all temperaturesgreater than 100°C.
Original inequality
Subtract 32 from each side.
Simplify.
Multiply each side by
Simplify.
Science The boiling point of helium is –452°F. Solve
the inequality to find the temperatures
in degrees Celsius for which helium is a gas.
Answer: Helium will be a gas for all temperatures greater than –268.9°C.
Then check your solution.
Original inequality
Subtract 13 from each side.
Simplify.
Divide each side by –11 andchange
Simplify.
Check To check the solution, substitute –6, a number less than –6, and a number greater than –6.
Answer: The solution set is
Then check your solution.
Answer:
Write an inequality for the sentence below. Then solve the inequality.
Four times a number plus twelve is less than a number minus three.
Four timesa number plus
is less than a numberminus three. twelve
4n + <12
Original inequality
Subtract n from each side.
Simplify.
Subtract 12 from each side.
Simplify.
Divide each side by 3.
Simplify.
Answer: The solution set is
Write an inequality for the sentence below. Then solve the inequality.
6 times a number is greater than 4 times the number minus 2.
Answer:
Original inequality
Add c to each side.Simplify.Subtract 6 from each side.Simplify.
Divide each side by 4.
Simplify.
Combine like terms.Distributive Property
Answer: Since is the same as
the solution set is
Answer:
Answer: Since the inequality results in a falsestatement, the solution set is the empty set Ø.
Original inequality
Distributive Property
Combine like terms.
Subtract 4s from each side.
This statement is false.
Answer: Ø
SOLVINGCOMPOUND INEQUALITIES
Example 1 Graph an Intersection
Example 2 Solve and Graph an Intersection
Example 3 Write and Graph a Compound Inequality
Example 4 Solve and Graph a Union
Graph the solution set of
Find the intersection.
Graph
Graph
Answer: The solution set is Note that the graph of includes the point 5. The graphof does not include 12.
Graph the solution set of and
Then graph the solution set.
First express using and. Then solve each inequality.
and
The solution set is the intersection of the two graphs.
Graph
Graph
Find the intersection.
Answer: The solution set is
Then graph the solution set.
Answer:
Travel A ski resort has several types of hotel rooms and several types of cabins. The hotel rooms cost at most $89 per night and the cabins cost at least $109 per night. Write and graph a compound inequality that describes the amount that a quest would pay per night at the resort.
Words The hotel rooms cost at most $89 per night and the cabins cost at least $109 per night.
Variables Let c be the cost of staying at the resort per night.
Inequality Cost pernight
is atmost $89 or
thecost
is atleast $109.
c 89 109cor
Now graph the solution set.
Graph
Graph
Find the union.
Answer:
Ticket Sales A professional hockey arena has seats available in the Lower Bowl level that cost at most $65 per seat. The arena also has seats available at the Club Level and above that cost at least $80 per seat. Write and graph a compound inequality that describes the amount a spectator would pay for a seat at the hockey game.
Answer: where c is the cost per seat
Then graph the solution set.
or
Graph
Graph
Answer:
Notice that the graph of contains every point in the graph of So, the union is the graph ofThe solution set is
Then graph the solution set.
Answer:
SOLVINGOPEN SENTENCES INVOLVINGABSOLUTE VALUE
Example 1 Solve an Absolute Value Equation
Example 2 Write an Absolute Value Equation
Example 3 Solve an Absolute Value Inequality (<)
Example 4 Solve an Absolute Value Inequality (>)
Method 1 Graphing
means that the distance between b and –6 is 5 units. To find b on the number line, start at –6 and move 5 units in either direction.
The distance from –6 to –11 is 5 units.
The distance from –6 to –1 is 5 units.
Answer: The solution set is
Method 2 Compound Sentence
Answer: The solution set is
Write as or
Original inequality
Subtract 6 from each side.
Case 1 Case 2
Simplify.
Answer: {12, –2}
Write an equation involving the absolute value for the graph.
Find the point that is the same distance from –4 as the distance from 6. The midpoint between –4 and 6 is 1.
The distance from 1 to –4 is 5 units.
The distance from 1 to 6 is 5 units.So, an equation is .
Check Substitute –4 and 6 into
Answer:
Write an equation involving the absolute value for the graph.
Answer:
Then graph the solution set.
Write as and
Original inequality
Add 3 to each side.
Simplify.
Case 1 Case 2
Answer: The solution set is
Then graph the solution set.
Answer:
Case 1 Case 2
Then graph the solution set.
Write as or
Add 3 to each side.
Simplify.
Original inequality
Divide each side by 3.
Simplify.
Answer: The solution set is
Then graph the solution set.
Answer:
GRAPHING INEQUALITIESIN TWO VARIABLES
Example 1 Ordered Pairs that Satisfy an Inequality
Example 2 Graph an Inequality
Example 3 Write and Solve an Inequality
From the set {(3, 3), (0, 2), (2, 4), (1, 0)}, which ordered pairs are part of the solution set for
Use a table to substitute the x and y values of each ordered pair into the inequality.
false01
true42
false20
true33
True or Falseyx
Answer: The ordered pairs {(3, 3), (2, 4)} are part of the solution set of . In the graph, notice the location of the two ordered pairs that are
solutions for in relation to the line.
From the set {(0, 2), (1, 3), (4, 17), (2, 1)}, which ordered pairs are part of the solution set for
Answer: {(1, 3), (2, 1)}
Step 1 Solve for y in terms of x.Original inequality
Add 4x to each side.
Simplify.
Divide each side by 2.
Simplify.
Step 2 GraphSince does not include values when the boundary is not included in the solution set. The boundary should be drawn as a dashed line.Step 3 Select a point in one of the half-planes and test it.Let’s use (0, 0).
Original inequality
false
y = 2x + 3
Answer: Since the statement is false, the half-plane containing the origin is not part of the solution. Shade the other half-plane.
y = 2x + 3
Answer: Since the statement is false, the half-plane containing the origin is not part of the solution. Shade the other half-plane.
Check Test the point in the other half-plane, for example, (–3, 1).
Original inequality
Since the statement is true, the half-plane containing (–3, 1) should be shaded. The graph of the solution is correct.
y = 2x + 3
Answer:
Journalism Lee Cooper writes and edits short articles for a local newspaper. It generally takes her an hour to write an article and about a half-hour to edit an article. If Lee works up to 8 hours a day, how many articles can she write and edit in one day?Step 1 Let x equal the number of articles Lee can write. Let y equal the number of articles that Lee can edit. Write an open sentence representing the situation.
Number of articles
she can write plus times
number of articles
she can edit is up to 8 hours.
hour
x + 8y
Step 2 Solve for y in terms of x.
Original inequality
Subtract x from each side.
Simplify.
Multiply each side by 2.
Simplify.
Step 3 Since the open sentence includes the equation,graph as a solid line. Test a point in one of the half-planes, for example, (0, 0). Shade the
half-plane containing (0, 0) since is true.
Answer:
Step 4 Examine the situation.
Lee cannot work a negative number of hours. Therefore, the domain and range contain only nonnegative numbers.
Lee only wants to count articles that are completely written or completely edited. Thus, only points in the half-plane whose x- and y- coordinates are whole numbers are
possible solutions.
One solution is (2, 3). This represents 2 written articles and 3 edited articles.
Food You offer to go to the local deli and pick up sandwiches for lunch. You have $30 to spend. Chicken sandwiches cost $3.00 each and tuna sandwiches are $1.50 each. How many sandwiches can you purchase for $30?
Answer:
The open sentence that represents this situation is where x is the number of chicken sandwiches,
and y is the number of tuna sandwiches. One solution is (4, 10). This means that you could purchase 4 chicken sandwiches and 10 tuna sandwiches.
THIS IS THE END
OF THE SESSION
BYE!