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Time to start another new section!!!
P3: Solving linear
equations and linear
inequalities
Algebraic Properties
Properties of Equality (let u, v, w, and z be real numbers,
variables, or algebraic expressions)
1. Reflexive
u=u
2. Symmetric
If u = v, then v = u
3. Transitive
If u = v, and v = w, then u = w
4. Addition
If u = v and w = z, then u + w = v + z
5. Multiplication
If u = v and w = z, then uw = vz
Solving Equations
A linear equation in x is one that can be written
in the form
ax + b = 0
where a and b are real numbers with a = 0
A solution of an equation in x is a value of x for
which the equation is true.
 So, how many solutions are there to a linear
equation in one variable???
Let’s practice… Solve for the
unknown:
2  2 x  3  3  x  1  5x  2
5
x   2.5
2
5y  2
y
 2
8
4
Let’s practice… Solve for the
unknown:
K.T.D.!!!
5y  2
y
8
8 (multiply both sides of the
 2
8
4
equation by the L.C.D.)
y6
Let’s practice… Solve for the
unknown and support with grapher:
4  4 z  2  z  3
1
z
3
Now, how do we get graphical support???
Definition: Linear Inequality in x
A linear inequality in x is one that can be written
in the form
ax + b < 0, ax + b < 0, ax + b > 0, or ax + b > 0
where a and b are real numbers with a = 0
A solution of an inequality in x is a value of x
for which the inequality is true.
The set of all solutions of an inequality is the
solution set of the inequality.
Properties of Inequalities
Let u, v, w, and z be real numbers, variables, or algebraic
expressions, and c a real number.
1. Transitive If u < v and v < w, then u < w
2. Addition
If u < v, then u + w < v + w
If u < v and w < z, then u + w < v + z
3. Multiplication
If u < v and c > 0, then uc < vc
If u < v and c < 0, then uc > vc
(the above properties are true for < as well – there are
similar properties for > and >)
Guided Practice:
Solve the inequality:
3  x 1  2  5x  6
7
x
2
Guided Practice:
Solve the inequality, write your answer in interval
notation, and graph the solution set:
x 1 x 1
  
3 2 4 3
x  2
( 2,  )
–2
0
Guided Practice
Solve the double inequality, write your answer in
interval notation, and graph the solution set:
2x  5
3 
5
3
7  x  5
(–7, 5]
–7
0
5
Whiteboard Practice:
Solve the inequality, write your answer in interval
notation, and graph the solution set:
3  m 5m  2

 1
2
3
11
m
7
, –
8
–
11
7
–11/7
0
Whiteboard Practice:
Solve the inequality:
7w  8  3w  2  4  7  2w
19
w
2
19
( , )
2
Whiteboard Practice: Solve and
support with grapher:
c 1 c  5 1


12
12
3
4
2
5
 c    0.714
7
Graphical Support?
Homework:
 p. 28-29 11-27 odd, 35-53 odd