6.1 Solving Inequalities in One Variables · –10 x–9 –8 –7 –6 –5 –4 –3 –2 –1 01...

6.1 Solving Inequalities in One Variables

Mr. Noyes, Akimel A-al Middle School 1 Heath Algebra 1 - An Integrated Approach

Objectives:

• Learn how to graph linear inequalities in one variable

• Learn how to solve linear inequalities in one variable

Graphing Inequalities: Phrase Inequality Graph

Less than x < 2 0 1 2 3 4 50–1–2–3–4–5 x

Greater than x > 5 0 1 2 3 4 5 6 7 80–1–2 x

Less than or equal to x ≤ 3 0 1 2 3 4 5 6 7 80–1–2 x

Greater than or equal to x ≥ –2 0 1 2 3 4 50–1–2–3–4–5 x

- open circle for >, < - closed circle for ≥, ≤

Summer temperature in Phoenix, T ≥ 80°

0 20 40 60 80 100 1200 T

Average snowfall is less than 1”

0 1 2 30–1–2–3 Can it go below zero?



Solving linear inequalities in one variable: Graph on a number line not coordinate plane (Cartesian plane)

We solve inequalities in one variable much like we solve equations in one variable. We need to isolate the variable on one side by using transformations.

However If you multiply or divide by a negative number - change the sign (reverse the direction) of the inequality

2x + 3 = 4 –3 –3 2x = 1

2 2

x = 12

2x + 3 > 4 –3 –3 2x > 1

2 2

x > 12

0 10–1 0 10–1 Graph (plot point) Graph (use open circle)

3 + x ≥ 9 –3 –3

x ≥ 6

x – 8 < –10 + 8 + 8

x < – 2

0 1 2 3 4 5 6 7 80 0 1 2 30–1–2–3–4–5

Graph (use closed circle) Graph (use open circle)



Solve and Graph!

– x – 8 > –17 + 8 + 8 – x > – 9

(–1) – x > – 9 (–1)

x < 9

x + 3 ≤ 2 (x – 4)

x + 3 ≤ 2x – 8 – x –x

3 ≤ x – 8 + 8 + 8

11 ≤ x

re–write x ≥ 11

0 1 2 3 4 5 6 7 8 9 100 x 5 6 7 8 9 10 11 12 13 14 15 x

Graph (use open circle) Graph (use closed circle) In Oswego, NY, the temperature in January may not exceed 0° C. Write an inequality that describes the temperature T for the month and graph it.

T ≤ 0

0 1 2 3 4 5 6 7 8 9 100–1–2–3–4–5–6–7–8–9–10 x

6.2 Problem Solving Using Inequalities


Objectives:

• Learn how to write and use a linear inequality as a model for a real-life situation

Example 1: You have $32.14 to spend, but sales tax is 6%, so how much can you spend so that you still have enough money to pay the sales tax?

Verbal Model:

cost + sales tax ≤ Total money to spend

Labels:

c = cost of items purchased

Algebraic Inequality:

c + 0.06c ≤ 32.14

1.06c ≤ 32.14 1.06 1.06

c ≤ 30.32

Example 2: If you go into business selling model cars for $18 and costs you $2,000 to set-up the

business and then $13 per car to produce them, how many cars must you sell to

realize a profit?

Verbal Model:

# of cars · 18 ≥ # of cars · 13 + set-up cost

6.2 Problem Solving Using Inequalities


Labels:

c = # of cars needed to sell in order to make a profit

set-up cost = $2,000

Algebraic Inequality:

18c ≥ 13c + 2000 –13c –13c

5c ≥ 2000 5 5

c ≥ 400 400 or more cars must be sold to make a profit Example 3: (See Guided Practice Problems 1 - 4 on page 301 of textbook)

You are placing an order for blank VCR tapes through the mail. Each tape costs $3,

including tax. Shipping and handling costs are $2 for the entire order. If you cannot

spend more than $20, how many tapes can you order?

1. Verbal Model:

Cost per tape · # of tapes + shipping cost ≤ $ to spend

2. Labels:

t = tapes ordered

3. Algebraic Inequality:

3t + 2 ≤ 20 4. Solve:

3t + 2 ≤ 20 – 2 – 2

3t ≤ 18 3 3

t ≤ 6

6.3 Compound Inequalities


Objectives:

• Learn how to solve and graph compound inequalities

• Learn how to model a real-life situation with a compound inequality

Compound Inequalities: are used to define or exclude a range of numbers

1. Between (Defining a range of numbers)

a < x < b or a ≤ x ≤ b Read as:

“x is between a and b” Read as:

“x is between a and b, inclusive”

Graph: 1 < x < 5

0 1 2 3 4 5 6 70–1 x 2. Greater than or Less than (excluding a range of numbers)

x < –1 x ≥ 2 We are excluding the numbers between –1 and 2 Graph:

0 1 2 3 40–1–2–3–4 x Solving and Graphing Compound Inequalities:

Example: –9 < 6x + 3 ≤ 39 –3 –3 –3

–12 < 6x ≤ 36 6 6 6 –2 < x ≤ 6

0 1 2 3 4 5 60–1–2 x

Notice that one point is open, and the other is closed

6.3 Compound Inequalities


Example:

8 + 2x < 6 –8 –8 2x < –2 2 2

x < –1

0 1 2 30–1–2–3 x

Example:

3x – 2 > 13 + 2 + 2 3x > 15 3 3

x > 5

1 2 3 4 5 6 7 x

Remember to reverse the inequality signs if you multiply or divide by a negative number.

Example:

Write a compound inequality for representing 4 misprinted stamps that originally sold

for 24¢ new in 1957 that were purchased in auction for $2,500 in 1996.

4 · (0.24) ≤ x ≤ 2500

.96 ≤ x ≤ 2500

6.4 Connections: Absolute Value and Inequalities


Objectives:

1. Learn how to solve absolute value inequalities

2. Learn how to model a real-life situation with an absolute value inequality

Properties of Absolute Value Inequalities:

1. The inequality |ax + b| < c is equivalent to

–c < ax + b < c

2. The inequality |ax + b| > c is equivalent to

ax + b < –c or ax + b > c Examples:

|x – 4| < 3 means –3 < x – 4 < 3 + 4 + 4 + 4

1 < x < 7

0 1 2 3 4 5 6 7 80 x

|x + 9| > 13 means

x + 9 > 13 or x + 9 < –13 – 9 – 9 – 9 – 9

x > 4 x < –22

0 2 4 60–2–4–6–8–10–12–14–16–18–20–22–24 x



Write Absolute Value Inequalities Write an absolute value inequality for the graph shown

0 1 2 3 4 5 6 7 8 90 x

Step 1: write the compound inequality for the graph

2 < x < 8 Step 2: find the number that lies halfway between 2 and 8. For this particular

question, this number is 5.

Step 3: subtract the number found in step 2 from each expression in the compound inequality, and re-write.

2 – 5 < x – 5 < 8 – 5

– 3 < x – 5 < 3 Step 4: re-write the compound inequality as an absolute inequality.

|x – 5| < 3 For this inequality, you can say that the range of solutions is “from 3 less than 5 to 3 greater than 5.”

–3 +3



Write an absolute value inequality for the graph shown

0 4 8 120–4–8–12–16 x Step 1: write the compound inequality for the graph

x < – 12 or x > 4

Step 2: find the number that lies halfway between – 12 and 4. For this question, this number is – 4.

Step 3: subtract the number found in step 2 from each expression in the compound inequality, and re-write.

x < – 12 or x > 4

x – (– 4) < – 12 – (– 4) x – (– 4) > 4 – (– 4)

x + 4 < – 8 x + 4 > 8 Step 4: re-write the compound inequality as an absolute inequality.

|x + 4| > 8

–8 +8

6.5 Graphing Linear Inequalities in Two Variables


Objectives:

• Learn how to graph a linear inequality in 2 variables

• Learn how to model real-life situations with linear inequalities in 2 variables

A Linear Inequality in x and y is an inequality that can be written in one of the following forms:

ax + by < c ax + by > c

ax + by ≤ c ax + by ≥ c An ordered pair (a, b) is a solution of a linear inequality when a and b are substituted for x and y, respectively. Example: (1, 3) is a solution of

4x – y < 2

4(1) – 3 < 2

4 – 3 < 2

1 < 2

True Checking solutions: are the ordered pairs (1, 3) and (2, 0) solutions to 3x – 2y < 2?

(1, 3) (2, 0)

3x – 2y < 2 3x – 2y < 2

3(1) – 2(3) < 2 3(2) – 2(0) < 2

3 – 6 < 2 6 – 0 < 2

-3 < 2 6 < 2

Yes NO



Graphing Inequalities:

Example: 3x – 2y < 2

First, re-write the inequality in slope-intercept form

3x – 2y < 2 – 3x – 3x – 2y < – 3x + 2 – 2 – 2 – 2

y > 32− x – 1 Remember, if multiplying or dividing by a

negative number, the inequality sign reverses! Graph the line as if it were a linear equation, however, draw the line according to the following rules:

If Less Than (<) or Greater Than (>) use a dashed line ( - - - - )

If Less Than or Equal To (≤) or Greater Than or Equal To (≥)

use a solid line ( _____ )

1 2 3 4 5 6 7–1–2–3–4–5–6–7–8 x

12345678

–1–2–3–4–5–6–7–8

y



Test a point on one side of the line to determine if it is a solution, and side to shade (shade true side) Helpful Tip! Pick point (0, 0) to Test 3x – 2y < 2 3(0) – 2(0) < 2 0 – 0 < 2 0 < 2 True Horizontal and Vertical Inequalities:

x < –3

1 2 3 4 5 6 7–1–2–3–4–5–6–7–8 x

12345678

–1–2–3–4–5–6–7–8

y

y ≥ 4

1 2 3 4 5 6 7–1–2–3–4–5–6–7–8 x

12345678

–1–2–3–4–5–6–7–8

y

(0, 0)

1 2 3 4 5 6 7–1–2–3–4–5–6–7–8 x

12345678

–1–2–3–4–5–6–7–8

y



Deep sea treasure hunters pull 50 pounds (lbs) of gold and silver coins from the sea. What proportions of coins could they have? Change pounds (lbs) to ounces (oz); 800 oz

50 lbs. · 16 oz.lbs. = 800 oz.

Each gold coin is 12 oz.

Each silver coin is 14 oz.

Let x = number of gold coins Let y = number of silver coins Ax + By ≤ C

21 x +

41 y ≤ 800

Find both the x- and y-intercepts: solve for y when x = 0 and solve for x when y = 0

41 y ≤ 800

y ≤ 3200

21 x ≤ 800

x ≤ 1600

400 800 1200 1600 Gold

800

1600

2400

3200

Silver

6.6 Exploring Data: Time Lines, Picture Graphs and Circle Graphs


Objective:

• Learn how to draw and interpret visual models such as time lines, picture graphs and circle graphs

Visual aids: Time Line: used to show special events in chronological time order (history)

Picture graphs: a stylized version of a bar graph, eye-catching, easy to read

6.6 Exploring Data: Time Lines, Picture Graphs and Circle Graphs


Circle graph: used to show relationships of relative sizes of different portions of a whole (i.e. budget, daily routine)

6.1 Solving Inequalities in One Variables · –10 x–9 –8 –7 –6 –5 –4 –3 –2 –1 01...

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