1-10 introduction to inequalities warm up warm up lesson presentation lesson presentation problem of...
TRANSCRIPT
1-10 Introduction to Inequalities
Warm UpWarm Up
Lesson PresentationLesson Presentation
Problem of the DayProblem of the Day
Lesson QuizzesLesson Quizzes
1-10 Introduction to Inequalities
Warm UpSolve.
1. x + 6 = 132. 8n = 483. t – 2 = 564. 6 =
x = 7
n = 6t = 58
z = 36z6
1-10 Introduction to Inequalities
Problem of the Day
Bill and Brad are taking Drivers Education class. Bill drives with his instructor for one and a half hours three times a week. He needs a total of 27 hours. Brad drives two times a week, two hours each time. He needs 26 hours. Who will finish his hours first? Bill
1-10 Introduction to Inequalities
Learn to solve and graph inequalities.
1-10 Introduction to Inequalities
Vocabularyinequalityalgebraic inequalitysolution set
1-10 Introduction to Inequalities
An inequality compares two quantities and typically uses one of these symbols:
<<is less than
is greater than
is less than or equal to
is greater than or equal to
1-10 Introduction to Inequalities
The inequality symbol opens to the side with the greater number.
2 < 10
Remember!
1-10 Introduction to Inequalities
Additional Example 1: Completing an Inequality
Compare. Write < or >.
A. 23 – 14 6
9 6>
B. 5(12) 70
60 70<
1-10 Introduction to Inequalities
Check It Out: Example 1
Compare. Write < or >.
A. 19 – 3 17
16 17<
B. 4(15) 50
60 50>
1-10 Introduction to Inequalities
An inequality that contains one or more variables is an algebraic inequality.
A number that makes an inequality true is a solution of the inequality.
The set of all solutions is called the solution set. The solution set can be shown by graphing it on a number line.
1-10 Introduction to Inequalities
x < 5
4 < 5x = 2.1 2.1 < 5
x is less than 5Word
Phrase
Inequality
Sample Solutions
Solution Set 1 2 3 4 5 6 7
x = 4
1-10 Introduction to Inequalities
a > 0
7 > 0a = 25 25 > 0
a is greater than 0
a is more than 0Word
Phrase
Inequality
Sample Solutions
Solution Set–3 –2 –1 0 1 2 3
a = 7
1-10 Introduction to Inequalities
y 2
0 2y = 1.5 1.5 2
y is less than or equal to 2
y is at most 2Word
Phrase
Inequality
Sample Solutions
Solution Set–3 –2 –1 0 1 2 3
y = 0
1-10 Introduction to Inequalities
m 3
17 3m = 3 3 3
m is greater than or equal to 3
m is at least 3Word
Phrase
Inequality
Sample Solutions
Solution Set–1 0 1 2 3 4 5
m = 17
1-10 Introduction to Inequalities
Most inequalities can be solved the same way equations are solved.
Use inverse operations on both sides of the inequality to isolate the variable.
1-10 Introduction to Inequalities
An open circle means that the corresponding value is not a solution. A solid circle means that the value is part of the solution set.
Helpful Hint!
1-10 Introduction to Inequalities
Additional Example 2A: Solving and Graphing Inequalities
Solve and graph the inequality.
x + 2.5 8 –2.5 –2.5
x 5.5
1 2 3 4 5 6 7
Use the Subtraction Property of Inequality:Subtract 2.5 from both sides.
According to the graph, 5.4 is a solution, since 5.4 < 5.5, and 6 should not be solution because 6 > 5.5.
1-10 Introduction to Inequalities
Additional Example 2A Continued
Check
Substitute 5.4 for x.
7.9< 8 ?
So 5.4 is a solution.
x + 2.5 < 8 ?
5.4 + 2.5 < 8 ?
Check
Substitute 6 for x.
8.5< 8 ?
So 6 is not a solution.
x + 2.5 < 8 ?
6 + 2.5 < 8 ?
1-10 Introduction to Inequalities
Additional Example 2B: Solving and Graphing Inequalities
Solve and graph the inequality.
w – 1 < 8
w < 9
–3 0 3 6 9 12 15
+ 1 + 1 Use the Addition Property of Inequality: Add 1 to both sides.
1-10 Introduction to Inequalities
Check It Out: Example 2
Solve and graph each inequality.
A. x + 2 3.5 –2 –2x 1.5
1 2 3 4 5 6 7
Use the Subtraction Property of Inequality: Subtract 2 from both sides.
B. 6u > 72
6 6
u > 12 3 6 9 12 15 18 21
6u > 72 Use the Division Property of Inequality: Divide both sides by 6.
1-10 Introduction to Inequalities
Standard Lesson Quiz
Lesson Quizzes
Lesson Quiz for Student Response Systems
1-10 Introduction to Inequalities
Lesson Quiz
Compare. Use < or > to compare each inequality.
1. 13 5(2) 2. 14 – 2 11
Solve and graph each inequality.
3. k + 9 < 12
4. m – 4 2
5. A school bus can hold no more than 64 passengers. There are already 21 passengers on the bus. Write and solve an inequality to find how many more passengers the bus can hold.
>
m 6
>
k < 3–5 –4–3–2–1 0 1 2 3 4 5
–4 –3–2–1 0 1 2 3 4 5 6
x + 21 64; x 43
1-10 Introduction to Inequalities
1. Identify the correct sign to compare.
19 4(4)
A. >
B. <
C. ≥
D. ≤
Lesson Quiz for Student Response Systems
1-10 Introduction to Inequalities
2. Identify the correct sign to compare.
19 – 3 21
A. >
B. <
C. ≤
D. ≥
Lesson Quiz for Student Response Systems
1-10 Introduction to Inequalities
3. Solve and graph the inequality.
x + 7 < 10
A. x < 3
B. x < 4
Lesson Quiz for Student Response Systems
0 1 2 3 4 5 6 7 8 9 100–1–2–3–4–5–6–7–8–9–10
0 1 2 3 4 5 6 7 8 9 100–1–2–3–4–5–6–7–8–9–10
1-10 Introduction to Inequalities
4. Solve and graph the inequality.
4 ≤
A. 8 ≤ a
B. 6 ≤ a
Lesson Quiz for Student Response Systems
a2
0 1 2 3 4 5 6 7 8 9 100–1–2–3–4–5–6–7–8–9–10
0 1 2 3 4 5 6 7 8 9 100–1–2–3–4–5–6–7–8–9–10
1-10 Introduction to Inequalities
5. An egg tray can hold 4 dozen eggs. Two friends have 24 eggs each. They would like to put all the eggs in the tray. Write and solve an inequality to determine whether all the eggs will fit in the tray.
A. 2(24) ≤ 4; no
B. 2(24) ≤ 48; yes
C. 2(48) ≤ 24; yes
D. 2(24) ≤ 24; no
Lesson Quiz for Student Response Systems