Download Linear Inequalities

Vocabulary and Signs
Linear Inequalities
Objective: To graph intervals and
solve inequalities.
Vocabulary and Signs
z Linear inequality – Ax + B < C
z Solution of an inequality- any number that
makes the inequality true.
g
z Signs
z < Less than
< Less than or equal to
z > Greater than > Greater than or equal to
z Graph - a number line indicating solutions of
any inequality.
Interval Notation
z Linear inequality: Ax + B < C
z Solution of an inequality- any number that
makes the inequality true.
g
z Signs
z < Less than
< Less than or equal
z > Greater than > Greater than or equal
z Graph: a number line indicating solutions
of any inequality.
zInterval notation: used to represent
solution sets
zopen interval: use parenthesis ( );
use with < or >; does not include
endpoint
zclosed interval: use brackets [ ]; use
with < or > ; includes endpoint
zNote: An interval can be half-open.
Graph each inequality. Write using
interval notation.
Interval Notation
Infinite intervals
z (a,∞) represents the set of real numbers
greater than a; {x | x > a}
(
(-----------------a
z (-∞,b] represents the set of real numbers
less than or equal to b; {x | x < b}
-------------------]
b
z (-∞,∞) represents all real numbers.
z A) x < -1
B) x > -3
-------------------)
-1
[------------------3
(−∞
∞, −1)
[3 ∞)
[3,
z C) -4[------------------)
<x<2
-4
2
(−4, 2)
1
Addition Property of Inequalities…
are the same as with equations…
Use the GOLDEN RULE
RULE.
Examples: Solve, check, and graph. Write
using interval notation.
zd) p + 6 < 8
p<2
(−∞
∞, 2)
zNote: Always write an inequality
with the variable on the left.
-------------------)
2
e) 8x < 7x – 6
x < -6
(−∞, −6)
-------------------)
-6
Multiplication Property
Solving with Multiplication
z Same as with equations with one
important difference:
If you multiply or divide by a
negative number
number, you must REVERSE
the inequality symbol.
Multiply both sides of each inequality by -5.
f) 7 < 8
g) -1 > -4
-35 < -40
5 > 20
Both of these statements are false. This
is why we MUST reverse the inequality
sign when multiplying or dividing by a
negative.
Examples: Solve, check and graph. Write
using interval notation.
Examples: Solve, check and graph.
Write using interval notation.
−m
j)
< −3
9
z h) 2x < -10
x < -5
(−∞, −5))
-------------------)
-5
i) -7k > 8
k≥
−8
7
−8 ⎞
⎛
∞, ⎟
⎜ −∞
7 ⎠
⎝
-2
[-----------------1
0
Multiply both sides by -9.
m > 27
(-----------------------------27
2
Examples: Solve, graph, and check.
Write using interval notation.
STEPS
1. Simplify. (Distribute, combine like terms,
eliminate fractions,…)
2 Get variable on one side by itself by
2.
performing the inverse operation.
3. Graph the solution on a number line.
4. Verify your solution.
Examples: Solve, graph, and check.
Write using interval notation.
z l) 5 – 3(m – 1) < 2(m + 3) + 1
5 – 3m + 3 < 2m + 6 + 1
-3m + 8 < 2m + 7
-5m < 1
1
⎡ 1 ⎤
m≥−
⎢⎣ − 5 , ∞ ⎥⎦
5
-1
[----------------------0
z k) 2k – 5 > 1 + k
k–5>1
k>6
[-----------------------------6
Examples: Solve, graph, and check.
1
3
m) (m + 3) + 2 ≤ (m + 8)
4
4
Multiply everything by LCD = 4.
⎡1
⎤
⎡3
⎤
4 ⎢ ( m + 3) ⎥ + 4(2) ≤ 4 ⎢ ( m + 8) ⎥
⎣4
⎦
⎣4
⎦
1(m + 3) + 8 ≤ 3(m + 8)
m + 3 + 8 ≤ 3m + 24
−2m + 11 ≤ 24
−2m ≤ 13
m≥−
13
2
⎡ −13 ⎤
⎢⎣ 2 , ∞ ⎥⎦
-7
[-----------------------6
Summary
z A solution of an inequality is any number
that makes the inequality true.
z Graphs
G h off inequalities
i
liti iindicate
di t allll possible
ibl
solutions.
3