Linear Equations and Inequalities

Objectives (1.2.1 and 1.2.2)

The students will be able to:

• solve linear equations and inequalities.

• determine the equation for a line.

• describe solutions using numbers, symbols, and/or graphs.

To solve two-step equations, “undo” the operations by working backwards.

Example:

Ask yourself:

1. What is the first thing we are doing to x?

2. What is the second thing?

Recall the order of operations as you

answer these questions.

• Dividing by 2

• Subtracting 3

3 72

x

To undo these steps, do the opposite operations in opposite order.

1. Draw “the river”2. Add 3 to both sides3. Simplify4. Clear the fraction -

Multiply both sides by 2

5. Simplify6. Check your answer

Use a DO-UNDO chart as a shortcut to answering the questions.

In the table, write the opposite operations in the opposite order

DO UNDO

÷2

-3

Follow the steps in the ‘undo’ column to isolate the variable.

3 72

x

+ 3 + 3

= - 4

x = -8

-4 – 3 = -7

2

x

+3

· 2

2 · · 2

38

72

1) Solve 2x - 1 = -3 + 1 + 1

2x = -2

2 2

x = -1

2(-1) - 1 = -3

-2 – 1 = -3

1. Draw “the river”2. Add 1 to both

sides3. Simplify4. Divide both sides

by 25. Simplify6. Check your

answer

D U

· 2- 1

+ 1÷ 2

2) Solve 3y – 1 = 8

7

3

A. y = 3

B. y = -3

C. y =

D. y = 7

3

Answer: A

3) Solve 16

3

d

A. d = -7

B. d = -19

C. d = -17

D. d = 17

Answer: B

Sample HSA Problem:

Alexis has $345 in her savings account and will deposit $15 each week. Alexis will not withdraw any money from her savings account.

After how many weeks will she have $1,035?

Answer: 46 weeks

What if there are variables on both sides of the equation?

1) Solve: 3x + 2 = 4x - 1

You need to get the variables on one side of the equation. It does not matter which variable you move. Try to move the one that will keep your variable positive.

1) Solve 3x + 2 = 4x - 1- 3x - 3x

2 = x - 1

+ 1 + 1

3 = x

3(3) + 2 = 4(3) - 1

9 + 2 = 12 - 1

1. Draw “the river”2. Subtract 3x from

both sides3. Simplify4. Add 1 to both

sides5. Simplify6. Check your

answer

A. -3

B.

C.

D. 3

What is the value of x if 3 - 4x = 18 + x?

1

3

1

3

Answer: A

3) Solve 4 = 7x - 3x

4 = 4x

4 4

1 = x

4 = 7(1) - 3(1)

1. Draw “the river”. – Notice the

variables are on the same side!

2. Combine like terms.3. Divide both sides by 4.4. Simplify.5. Check your answer.

What is the value of x if -3 + 12x = 12x - 3?

A. 0

B. 4

C. No solutions

D. Infinite solutions

Answer: D

Solving an Inequality:

• Solve 5z + 16 < 51

A. z < -35

B. z < -7

C. z < 35

D. z < 7

Answer: D

Solve: -7(x - 3) < -71. Draw “the river”.2. Distribute -7.3. Subtract 21 from both

sides. 4. Simplify.5. Divide both sides by -7.6. Simplify.7. Check your answer.

-7x + 21 < -7 - 21 - 21 -7x < -28 -7 -7 x > 4 Note: >

WHY is < reversed???

.

42 x

-4 -2 0 2 4

-4 -2 0 2 4

-4 -2 0 2 4

-4 -2 0 2 4

Which of these graphs represent the solution set of the inequality

A

B

C

D

Let’s look at #14 in your packet:

?

Answer: B

Definition of Slope

Definition of SLOPE: the ratio of the vertical and horizontal distances between any two points on a line.

Informally you will see some of these:

read as "rise over run"

RiseRun

1 2

1 2

y ym

x x

xy

the slope = the rate of change

Types of Slope

POSITIVE NEGATIVE

Types of Slope

ZERO NO slope

Counting the slope, given a graph

Step 1: find 2 points Step 2: up=pos down = neg right=pos left= neg

(4, -2)

(-2,8)

+ 10

-6

Step 3:

Slope = ____ 10

-6

Simplify = ____ 5

-3

Counting the slope, given a graph

No need to go further – Zero slope

No need to go further –

No slope

Finding the slope, given 2 points

Remember our formula? 1 2

1 2

y ym

x x

(2, 3) and (4, -1)

2 4

3 -1

(x1, y1) (x2, y2)

(2, 3) and (4, -1) m = ___

-2

4 -21

=

Don’t forget to simplify!

Finding the slope, given 2 points

Remember our formula? 1 2

1 2

y ym

x x

(3,6) and (3, 4)

3 3

6 4

(x1, y1) (x2, y2)

(3,6) and (3, 4) m = ___

0

2 No slope =

Remember when 0 is in the denominator

Slope as a rate of change

Sample Problem:

In 1990, a company had a profit of $1,300,000. In 1992, the company had a profit of $1,200,000. Find the average rate of change in dollars per year.

A rate of change is a ratio like the slope.

1990 1992

1,300,000 1,200,000dollars

year

ex: miles per hour ► mi/hrpeople per year ► people/year

per is represented by the fraction bar /

-

-=

-2

100,000

- 50,000 dollars/yr=

=

Practice:

Positive

Zero

No

Negative

What type of slope is this?

Example #2:58

- 8 5

85

- 5 8

What is the slope?

47

74

65

56

Determine the slope of the line through the

points

(2, 0) and (-4, -5).

Slope-Intercept Form:

y=mx+b

slope y-intercept

Q: What is the y-intercept?A: the point where the line crosses the y-

axis

Example:

The line crosses the y-axis at -2. So the

y-intercept is the point (0,-2).

If we have the y-intercept and we can calculate the slope,

then we can write the equation of the line.

y=mx+b

Put in slope Put in y-intercept

Write the equation of the line.

1

2

1)Find the y-intercept:

b = 1

2)Find the slope:

Let’s use the points

(0,1) and (2,0) to find the slope

m = -1/2

3) Write the equation:1

12

y x

Find the equation of the line.

1

2

1)Find the y-intercept:

b = 4

2)Find the slope:

Let’s use the points

(0,4) and (-4,0) to find the slope

m = 1/1

3) Write the equation:1

41

y x

Write the equation of the line.

1

2

1)Find the y-intercept:

b = -3

2)Find the slope:

The line has zero slope; m = 0

3) Write the equation:

0 3y x or

3y

Write the equation of the line.

1

2

1)Find the y-intercept:

There is none!

2)Find the slope:

The slope is undefined.

3) Write the equation:2x

Point to ponder:

Why do you think this is the equation form instead of y=?

Locating the Intercepts without a Graph:

Another way to describe the x and y-intercepts is to think about them as coordinate points on the graph.

x-intercept: ( ___, 0) y is always 0.

y-intercept: ( 0, ___)

x is always 0.

Without graphing, describe what the graph of the equation would

look like. Include information about the slope and y-intercept.

1) y = -5x+3 2) y = -5

Negative slope

m = -5

y-intercept (0,3)

Zero slope

m = 0

y-intercept (0,-5)

What if you knew 2 points on a line?

What extra step would you have to do to write the equation of the

line?

Answer: First find the slope, then substitute the coordinates of one

point for x and y.

Example: Find the equation of the line that contains these points:

(3, -6),(-5, 2) Slope =

Now pick a point to use and solve for b.(Do you think it matters which

point we use? Why not?)

y = -1x - 3

y = mx + b

y = -1x + b

6 2 81

3 ( 5) 8m

-6 = -1(3) + b

-3 = b

You are correct !!

You are correct !!

You are correct !!

Incorrect. Please try again.

The End

• (But lines do not end!!!!!)