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P-3
Linear Equations and Inequalities
Vocabulary
Linear Equation in one variable.
Ax + B = C
A≠0
B and C are constants
You’ve seen this before!
4x – 2 = 6
+ 2 +2
4x = 8
÷ 4 ÷4
x = 2
Give me another example of a
linear equation in one variable.
Linear Equation in 2 Variables
Ax + By = C
3x + 4y = 12
Linear Equation in 3 Variables
Ax + By + Cz = D
3x + 4y + 6z = 12
There is no limit to the number of variables in
a linear equation. Animation uses over 100.
What makes it a ‘linear” equation ?
If the exponent of the variable(s) is a ‘1’,
then it is a linear equation.
3x + 4y + 6z = 12
Solutions of Linear equations
What does “solution” mean ?
Vocabulary
Solution: the number the variable must equal
in order to make the statement true.
x+1=2
The solution is: x = 1
“Solving” Linear equations
What does it mean to “solve” an equation?
Vocabulary
Solve: using properties to re-write the equation
in the form: x = (some exact value)
in order to make the statement true.
x+1=2
-1
-1
x = 1
(subtraction property of equality)
(solution)
Distributive Property of Addition over
Multiplication
2(x + 4) = (2 * x) + (2 * 4)
= 2x + 8
Biggest 2 errors in the distributive property:
Trying to multiply when the operation is add or subtract
Failing to distribute a negative to BOTH terms inside parentheses
5 - (x - 4)
= 5 x – 20
5 - (x - 4)
=5–x–8
NO NO NO!!!
NO NO NO!!!
Your turn: Solve these equations.
3.
3
x
2
1.

4 3
2.
5x
1  3
7
4.
2x  7  3  4x
2x  3
 2  3x  5
5
Vocabulary
Linear Inequality (in one variable):
Ax + B < C or
3x – 2 < 4
Ax + B > C or
Ax + B ≤ C or
Ax + B ≥ C.
More Vocabulary
Equivalent Equations: An equation that has the
same solution as the original equation.
x+2=4
-2 -2
x=2
(subtraction property of equality)
Equivalent Inequalities: An inequality that has
the same solution as the original inequality.
x+2<4
-2 -2
x<2
(subtraction property of inequality)
Solving inequalities (variable on both sides
of a single inequality symbol)
3x + 1 ≤ 2x + 6
-2x
-2x
x+1 ≤
6
-1
-1
x≤5
KEY POINT: collect variable
on the side that will result in
a positive coefficient.
Your turn: Solve the inequality
≤
6
5.
2x + 2
6.
2(x – 3 ) ≥
7.
-14x – 2
8
< 5x + 6
The “Gotcha” of Inequalities
2 – 2x
+ 2x
≤
6
2 – 2x
+ 2x
-2
2
≤ 2x + 6
-6
-6
-4
≤ 2x
÷2
÷2
-2
≤ x
≤
6
-2
-2x
≤
÷ -2
x
4
÷ -2
≤
-2
Anytime you multiply or divide by a
negative number, you must
switch the direction of the inequality !!
Solving inequalities (variable on both sides
of a single inequality symbol)
3x + 1 ≤ 2x + 6
-2x
-2x
x+1 ≤
6
-1
-1
x≤5
To avoid the “gotcha”:
collect the variable
on the side that will result in
a positive coefficient.
Your turn: Solve the inequality
8.
2x – 6 ≤ 3 – x
9.
18 + 2x ≥ 9x + 4
10.
2(x – 4) < 4x + 6
Compound inequalities (two inequality symbols)
5 ≤ x+1< 9
5 ≤ x+1
-1
4 ≤ x
Same as:
and
x+1< 9
-1
-1
and
4 ≤ x < 8
-1
x < 8
Compound inequalities (two inequality
symbols)
5 ≤ x+1< 9
-1
4 ≤ x
-1
-1
KEY POINT: subtraction
property of inequality  do
the same thing (left-middle-right)
< 8
Same as: 4 ≤ x and x < 8
Your turn: Solve the inequality
11.
-3 < 4 – x ≤ 3
12.
-5 < x + 1
and x + 1 ≤ 6
Solving inequalities (“or” type)
KEY POINT: treat “or” type compound
inequalities as two separate inequalities.
x-2 ≤ 3
+2 +2
or
x
≤ 5
or
x
≤ 5
or
x+2> 8
-2 -2
x >
x >
6
6
Your turn: Solve the inequality
13.
14.
4x - 7 ≤ 5 or 3x + 2 > 23
x + 1 ≤ -3 or x – 2 > 0
Sometimes there is no solution
Solution: the value(s) of the variable that make
the statement true.
2(x – 4) > 2x + 1
2x – 8
-2x
–8
> 2x + 1
-2x
> 1
No solution: when the
variable dissappears
and the resulting
statement is false.
Sometimes the solution is
all real numbers.
Solution: the value(s) of the variable that make
the statement true.
4x – 5
4x – 5
-4x
≤ 4(x + 2)
≤ 4x + 8
-4x
–5
≤ 8
Infinitely many solutions:
when the variable
dissappears and the
resulting statement is
true.
Your turn: Solve the inequality
15.
2(3x – 1) > 3(2x + 3)
16.
2x + 3 ≤ 3(x + 2) – x
Graphing Single Variable inequalities
x>3
What part of the number line is greater than 3 ?
1
2
3
4
5
Graphing Single Variable inequalities
x<5
What part of the number line is less than 5 ?
1
2
3
4
5
Your turn: Graph the following
17.
x≥7
18.
3>x
Graphing Compound Inequalities
x>3
and
x<5
And means both conditions must be met
What part is
x>3?
What part is
x < 5?
What is the intersection or overlap of the two?
1
2
3
4
5
Vocabulary
Compound inequality
x>3
and
Hint: This can also be written as:
3<x<5
Hint: Inequality with “and” looks like:
1
2
3
4
5
x<5

Your turn:
19.
20.
x>2
Graph the following compound inequalities.
and
-2 < x ≤ 5
x<6
Graphing “or” type compound inequalities.
x≤3
or
x>5
Or means: the points that satisfy either condition
Which part is x ≤ 3 ?
1
2
3
Which part is x > 5 ?
4
5
Hint: inequality with “OR” looks like:
 
Your turn:
21. Solve and graph the compound inequality:
2x + 3 ≤ 5 or x - 3 > 2
1
2
3
4
5
Verbal Inequalities
The cost of a car is at most $20,000.
It takes Jehah no less than 5 minutes to run a mile.
It takes between 3 and 8 months to build a house.
The cost of a loaf of bread is less than $2
You can’t buy a car for less than $8000.
Your turn:
22.
23.
(a) Write in inequality notation
(b) Graph the inequality
There are least 65,000 spectators at the game.
It never gets above 100 degrees in Huntsville.
24. You can fit, at most, 5 cars in your garage.
Three Ways to show an Inequality
x≤2
x>3
1. Inequality:
2. Bracket Notation: (3,  )
3. Number line Notation:
1
2
3
4
5 6
(  , 2]
Inequalities Involving Fractions
x 1 x 1
  
3 2 4 3
Another example?
x5 x2 1


8
2
3
Your Turn: Solve this inequality
2 x  3 3x  1 x  1
25.


2
5
3
Homework
P-3: evens: 2-10, 18-26, 32-44, 54
(18 problems)