solving linear equations and inequalities solving algebraically solving graphically solving...
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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