Solving Linear Equations and

Inequalities

• Solving algebraically

• Solving graphically

• Solving equations in more than one variable

• Solving linear inequalities

• Solving double inequalities

• Solving absolute value equations

• Applications

• Solving AlgebraicallyExample: Solve 2x = x – 92x – x = x – 9 – x

(get x on 1 side)x = -9 (simplify)

• Solving GraphicallyGraph left hand side of equation

and right hand side of the equation

and see where the graphs meet.

x = -9 (just need x-value of point)

y = 2x and y = x -9

• Solve 2x + 1 = -x –9

2x + 1 + x = -x –9 + x (x on 1 side)

3x + 1 = -9 (simplify)

3x + 1 – 1 = - 9 – 1 (get x on own)

3x = -10 (simplify)

x = -10

3

Check: Replace x with answer.

2(-10/3) + 1 = -(-10/3) – 9

-20/3 + 1 = 10/3 – 9

-17/3 = -17/3

Both sides equal so answer is correct.

• Graph both sides

x = -3.34 (no y-value required)

• Solve 2x = 2x – 3

2x – 2x = 2x – 3 – 2x

0 = -3 (impossible)

So no answer!

• Graph

Lines coincide, so they intersect everywhere. (same line)

•Solve 2(x + 1)=2x + 2

2x + 2 = 2x + 2 (distribute)

2x + 2 – 2x = 2x + 2 – 2x

2 = 2 (always true)

So every x is an answer!

Lines parallel so no intersection

Example: Solve algebraically.

x

x

x

x

x

xxxx

x x xx

x

xx

7

177

7

7

17

simplify 717

own sit'on get x 24247247

simplify 2477

side oneon sget x' 624667

simplify 2467

LCDby multiply )12

2

13

2

7(2

12

2

13

2

7

Solve: xx

12

2

13

2

7

x = -2.5

Words of caution:

Solving graphically will give you answers that are imprecise. If you want accuracy you need to solve algebraically.

If I ask you to solve an equation I want a precise answer. However, you can see if you are in the ball park by graphing.

Solving equations in more than one variable

9

)32(59

)32(55

9

9

5

9

5)32(

5

932

32325

932

325

9

FC

CF

CF

CF

CF

CFSolve for C

Solving Linear Inequalities

A linear inequality is similar to a linear equation except it is

an inequality. Here are some examples of linear inequalities.

)2(4432

12

32)5(25

332

xx

xx

xxx

x

Solved 14

own sit'on Get x 9599

Simplify 59

side 1on sGet x' 252293

Simplify 5293

52)3(3

x

x

x

xxxx

xx

xxSolve the linear inequality and graph on a number line.

-14

(-, -14) in interval notation

3

inequality flip negative aby divided 3

9

3

3

93

211223

1123

x

x

x

x

x

Solve the inequality and graph the solution on the number line.

-3

( -, -3 ] in interval notation

3

24

inequality flip so negativeby divided 3

14

143

593

5)279(3

1

x

x

x

x

x

Solve the inequality and graph solution on number line.

3

24

[ -4 2/3, ) in interval notation

Solving a Double Inequality

part.every fromaway 5 take weso 2010

itselfby get x to trying5255555

2555

x

x

x

Solve and graph on a number line.

-10 20

( -10, 20 ]

Solving Absolute Value Equations

Almost everyone has a hard time with these equations. The most common error is to only give one solution. When in fact there are usually two answers. Let’s try solving these graphically first. We will graph the left and right hand sides of the equation and see where the graphs meet.

Solve this absolute value equation.

5x

5 and yxy

y = | 5 | y = 5

x = - 5x = 5

Because the graph of an absolute value function is generally a ‘V’, there is a good chance that you will get two answers.

Solving Absolute Value Equations Algebraically

7

7or 7

pieces twointoequation Break

7

piece - The piece The

x

xx

x

Solve the absolute value equation.

1

22 11

12102 222

12)102( 12102

12102

piece - piece

x

xx

xx

xx

x

t

t

tt

tttt

tttt

tt

7

15

7t15 1

976 33

9265 325

92)65( 9265

9265

piece - piece

Solve the following inequality

Solving Absolute Inequalities Graphically

51 xSolve the absolute inequality graphically.

The absolute value function is larger than

y = 5 when x is >= 4 and x <= -6

or ( - , -6 ] [ 4, )

-6 4

xx 42

Solve the absolute inequality graphically

The absolute value function is less than the x function when x is less than 4 and greater than 4/3

4/3 4 or ( 4/3, 4 )

x 4/3

x = 4y = | 2x – 4 |

y = x

Now let’s solve algebraically

13

13

58 3

5)8( 58

piece - piece

58

x

x

xx

xx

x

So x must be less than –3 and greater than -13

-13 -3(-13, -3)

3

5

53

053 5

52 05

)52( 52

piece- piece

52

x

x

xx

xxx

xxxx

xxSolve

Answer in interval notation:

( - , -5/3 ] [ 5, ]

or 5/3 5

Applications

Break Even Analysis

I have decided to go into the business of making custom made tile top tables. The fixed cost of setting up my business is $1,000. Each table costs $75 to make. I plan on selling the tables for $115 each. How many tables will I need to sell in order to break even?

Solution: There are two functions here. A cost function C(x) and a revenue function R(x) (where x is number of tables sold.

C(x) = 1000 + 75x and R(x) = 115x

The break even point is the point where cost and revenue are equal. So set the functions equal to each other and solve.

1000 + 75x = 115x

1000 + 75x – 75x = 115x – 75x

1000 = 40x

1000 ÷ 40 = 40x ÷ 40

25 = x

So I will have to sell at least 25 tables to break even.

More on solving graphically

1. f(x) = g(x) 2. f(x) – g(x) = 0

3. f(x) > 0 4. f(x) > g(x)

5. g(x) < 0 6. f(x) < g(x)

ANSWERS

a. x=-5 & 1 & 5b. (- , -5] [ 1, 5 ]c. (-5,1) (5, )d. (-6,-2) (2, )e. [-4,1] [6,)

Solutions:

1-a 2-a 3-e4-c 5-d 6-b