Linear Equations and Inequalities

Much education today is monumentally ineffective. All too often we are giving young people cut flowers when we should be teaching them to grow their own plants. John W. Gardner

Algebra is great fun - you get to solve puzzles!

A Puzzle What is the missing number?

– 2 = 4

Algebra

Well, in Algebra we don't use blank boxes, we use a letter (usually an x or y, but any letter is fine). So we would write:

x

– 2 = 4

x

– 2 = 4

It is really that simple. The letter (in this case an x) just means "we don't know this yet", and is often called the unknown or the variable.

And when you solve it you write:

x

= 6

How to Solve

Algebra is just like a puzzle where you start with something like: x – 2 = 4 and you want to end up with something like: x = 6

How to Solve

Basic approach to solve:  Work out what to remove to get "x = ..."  Remove it by doing the opposite (adding is the opposite of subtracting)  Do that to both sides

How to Solve

To remove it, do the opposite, in this case add 2: Do it to both sides: Which is: We want to remove the “-2"

x

– 2 = 4

x

– + 2 0 0 + 2 6 = 6 x = 6

Why did we add 2 to both sides?

To "keep the balance“ Add 2 to left side Add 2 to right side Out of balance Back in balance

Algebra-Balance Scale

Solve simple linear equations using a balance beam representation.

3x + 3 = x + 7

Algebra-Balance Scale

Add tiles representing variables and constants to both sides of the scale to balance the equality.

3x + 3 = x + 7

Algebra-Balance Scale

What happens if we move a number 1 from left hand side

3x + 3 = x + 7

No longer balanced……… 3x + 2 does NOT equal x + 7 What do we need to do to make it balance?

Algebra-Balance Scale

We need to move 1 from the other side to get a new balance equation

3x + 2 = x + 6

We have balance again since 3x + 2 does EQUAL x + 6

Algebra-Balance Scale

Let’s remove 2 from both sides to eliminate the constant from one side of the equation

3x = x + 4

We don’t have numbers on both sides; but we do have x’s

Algebra-Balance Scale

Now let’s remove the “x” from the right hand side of the equation

3x = x + 4

Taking x just from one side caused imbalance We must take x from other side

Algebra-Balance Scale

Now we have balance with 2x on one side and 4 on the other side

2x = 4

Since twice a number is 4, then x = 2 ………….wow easy

Algebra-Balance Scale 3x + 3 = x + 7 3x + 3 - 3 = x + 7 - 3 3x = x + 4 3x -x = x - x + 4 2x = 4 x = 2

Subtract 3 from both sides Subtract x from both sides Divide by 2 from both sides

Solve: r + 16 = -7

To solve, you must get the variable by itself. What number is on the same side as r ?

16

To get r by itself, we must undo the “add 16”.

What is the opposite of addition?

Subtract 16

1.

2.

3.

4.

Draw “the river” to separate the equation into 2 sides Subtract 16 from both sides Simplify vertically Check your answer by substituting your answer back into the problem

r + 16 = -7 - 16 -16 r = -23 -23 + 16 = -7

Solve:

x

+ 2 = -3 Get the variable by itself. What is your first step?

1.

2.

3.

4.

Add 2 to both sides Subtract 2 from both sides Add 3 to both sides Subtract 3 from both sides

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

Example:

x

2 7 Ask yourself, 1.

2.

What is the first thing we are doing to x?

What is the second thing?

• • Recall the order of operations as you answer these questions.

Dividing by 2 Subtracting 3 To undo these steps, do the opposite operations in opposite order.

Use a DO-UNDO chart as a shortcut to answering the questions. In the table, write the opposite operations in the opposite order

x

7 DO UNDO 2 ÷ 2 1.

Draw “the river” + 3 + 3 - 3 2 2.

3.

4.

Add 3 to both sides Simplify Clear the fraction -Multiply both sides by 2

x

2 × 2 x

= =

-8 Follow the steps in the ‘undo’ column to 5.

6.

Simplify Check your answer  2 8 - 4 × 2 7 isolate the variable.

-4 – 3 = -7

Solve:

1.

2.

3.

4.

5.

6.

Draw “the river” Add 1 to both sides Simplify Divide both sides by 2 Simplify Check your answer

2x - 1 = -3 + 1 + 1 2x = -2 2 2 x = -1 2(-1) - 1 = -3 -2 – 1 = -3

D U × 2 - 1 + 1 ÷ 2

1.

2.

3.

4.

5.

6.

Solve

Draw “the river” Clear the fraction - Multiply both sides by 2 2

d

 4  3

d

2  4  3 2 2

d – 4 = 6

Simplify

+ 4 + 4

Add 4 to both sides

d = 10

Simplify Check your answer 10  4 2  3 D U - 4 ÷ 2 × + 4 2

Solve

1.

2.

3.

4.

d = -7 d = -19 d = -17 d = 17

d

 1   6 3

Equivalent Equations

Equations with the same solutions are called

equations.

equivalent

The Addition Principle

For any real numbers

a, b,

and

c

,

a = b

is equivalent to

a + c = b + c.

The Multiplication Principle

For any real numbers

a, b,

and

c

with

c

 0,

a = b

is equivalent to

a • c = b • c.

Solve: Solution

Solve: 2

x

3 2 3

x

11 2 3

x

 5  5  11  5 2

x

3 2  3 2 3

x

 16  1 6  3 2  3 1

x

 24 2 11 Adding 5 to both sides (Addition Principle) Multiplying both sides by 3/2 (Multiplication Principle) Simplifying

Solve.

4 5 (6

x

Solution:

4 (6

x

5 5 4  4 5 (6

x

6

x

1) 8 5 4 6

x

 1  1 6

x

 10  9  1 6

x

6  9 6

x

 3 2 8

An Equation-Solving Procedure

1.

Use the multiplication principle to clear any fractions or decimals. (This is optional, but can ease computations.

2.

If necessary, use the distributive law to remove parentheses. Then combine like terms on each side.

3

. Use the addition principle, as needed, to isolate all variable terms on one side. Then combine like terms.

4.

Multiply or divide to solve for the variable, using the multiplication principle.

5.

Check all possible solutions in the original equation.

Problem Solving Strategy

1.

Familiarize

yourself with the problem (draw pictures if applicable).

2.

Translate

to mathematical language. (This often means writing an equation.)

3.

Carry out

some mathematical manipulation. (This often means

solving

an equation.)

4.

Check

your possible answer in the original problem.

5.

State

the answer clearly, using a complete English sentence.

The apartments in Wanda’s apartment house are consecutively numbered on each floor. The sum of her number and her next door neighbor’s number is 723. What are the two numbers?

Solution

1.

Familiarize.

The apartment numbers are consecutive integers. Let

x

= Wanda’s apartment Let

x

+ 1 = neighbor’s apartment

2.

Translate

.

Rewording

:

Translating

: First integer plus second integer is 723    

x

     723 3. Carry out. x + (

x

+ 1) = 723 2

x

+ 1 = 723 2

x

= 722

x

= 361 If

x

is 361, then

x

+ 1 is 362.

4.

Check

. Our possible answers are 361 and 362. These are consecutive integers and the sum is 723.

5.

State.

The apartment numbers are 361 and 362.

Digicon prints digital photos for $0.12 each plus $3.29 shipping and handling. Your weekly budget for the school yearbook is $22.00. How many prints can you have made if you have $22.00?

Solution

1.

Familiarize

. Suppose the yearbook staff takes 100 digital photos. Then the cost to print them would be the shipping charge plus $0.12 times 100.

$3.29 + $0.12(100) which is $15.29. Our guess of 100 is too small, but we have familiarized ourselves with the way in which the calculation is made.

2.

Translate

.

Rewording

: Shipping plus photo cost is $22

Translating

:  $3.29       22 3.

Carry out.

x

 22 0.12

x

 18.71

x

 4.

Check

. Check in the original problem. $3.29 + 155(0.12) = $21.89, which is less than $22.00.

5.

State.

week.

The yearbook staff can have 155 photos printed per

Solving Inequalities

● Solving inequalities follows the same procedures as solving equations.

● There are a few special things to consider with inequalities: ● We need to look carefully at the inequality sign.

● We also need to graph the solution set.

How to graph the solutions

< Graph any number

less than

. . .

open circle, line to the left  Graph any number

greater than or equal to

. . .

closed circle, line to the right  > Graph any number

greater than. . .

open circle, line to the right Graph any number

less than or equal to

. . .

closed circle, line to the left

Solutions of Inequalities

An inequality is a number sentence containing > (is greater than), < (is less than),  (is greater than or equal to), or  (is less than or equal to).

Determine whether the given number is a solution of

x

< 5: a)  4 b) 6

Solution

-3 -2 -1 0 1 2 3 4 5 6 a) b) Since  4 < 5 is true,  4 is a solution.

Since 6 < 5 is false, 6 is not a solution.

Graph each inequality: a)

x

c)  3 <

x

 5 < 3, b)

y

  4;

Solution

a) The solutions of

x

than 3.

< 3 are those numbers less Shade all points to the left of 3.

The open dot at 3 and the shading to the left indicate that 3 is

not

part of the graph, but numbers like 1 and  2 are. -3 -2 -1 0 1 2 3 4 5 6

b) The solutions of

y

  4 are shown on the number line by shading the point for  4 and all points to the right of  4. The closed dot at  4 indicates that  4 is part of the graph.

-6 -5 -4 -3 -2 -1 0 1 2 3 c) The inequality  3 <

x

 5 is read “  3 is less than

x and x

is less than or equal to 5.” -4 -3 -2 -1 0 1 2 3 4 5

The Addition Principle for Inequalities

For any real numbers

a, b,

and

c

:

a a

<

b



b

is equivalent to

a + c < b + c;

is equivalent to

a + c



b + c; a a

>

b



b

is equivalent to

a + c > b + c;

is equivalent to

a + c



b + c.

Solve

x

+ 6 > 2 and then graph the solution.

Solution

x x

+ 6 > 2 + 6 

x

6 > 2  >  4 6 -4 -3 -2 -1 0 1 2 3 4 5 Any number greater than  4 makes the statement true.

Solve 4

x

 1 

x

 10

Solution

4 4

x x

  1 

x

1 + 1  

x

10  4

x

 4

x x

 

x x

  9

x

10 + 1  9 3

x x

  9   3 Adding 1 to both sides Simplifying Subtracting

x

from both sides Simplifying Dividing both sides by 3 -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2

The Multiplication Principle for Inequalities

For any real numbers

a

number

c

: and

b

, and for any

positive a

<

b

is equivalent to

ac

<

bc

, and

a

>

b

is equivalent to

ac

>

bc

.

For any real numbers

a

number

c

: and

b

, and for any

negative a

<

b

is equivalent to

ac

>

bc

, and

a

>

b

is equivalent to

ac

<

bc

.

Similar statements hold for  and  .

.

Solve and graph each inequality: a) b) 7

x

  4

y

< 20

Solution

a) 1 7

x

 4 4  1 3

x x

 28 4 Multiplying both sides by 4 Simplifying 5 10 15 20 25

b)  4

y

< 20  4

y

 4  20  4

y

  5 Dividing both sides by  4

At this step,

we reverse the inequality, because  4 is negative.

-5 -4 -3 -2 -1 0 1 2 3 4