Download Solving and graphing MultiStepInequalities

Objective
The student will be able to:
1. graph inequalities on a number line.
2. solve inequalities using addition and
subtraction.
Designed by Skip Tyler, Varina High School
1) Graph the solution set of
x < 3.
o
-5
0
5
When you have < or >, use an open dot!
2) Graph the solution set of
y ≥ -5/4.
-5
•
0
5
When you have ≤ or ≥, use a closed dot!
Converting -5/4 to a decimal = -1.25
3) Graph the solution set of
p ≠ 2.
-5
0
o
5
When you have ≠, use an open dot and shade
both ways!
4) Which inequality would have a
closed dot on the number line?
1.
2.
3.
4.
>
<
≥
≠
Answer Now
5) Which inequality does NOT use an
open dot on the number line?
1.
2.
3.
4.
≤
<
>
≠
Answer Now
6) Solve x + (-14) < 16
x - 14 < 16
+ 14 + 14
x < 30
Solve this problem like
an equation
1. Draw “the river”
2. Eliminate double
signs
3. Add 14 to both sides
4. Simplify
5. Check your answer
6. Graph the solution
29
30 + (-14) = 16
16 = 16
o
30
31
1.
2.
7) Solve y + 21 ≥ 7
- 21 -21
Draw the “river”
Subtract 21 from
y
≥
-14
both sides
3. Simplify
4. Check your answer
5. Graph the solution
-15
(-14) + 21 = 7
7=7
●
-14
-13
8) Solve
1.
2.
3.
4.
5.
6.
7.
8.
Draw “the river”
Subtract 8y from both
sides
Simplify
Add 14 to both sides
Simplify
Rewrite inequality
with the variable first
Check your answer
Graph the solution
16
8y + 3 > 9y - 14
- 8y
- 8y
3 > y - 14
+ 14 + 14
17 > y
y < 17
8(17) + 3 = 9(17) - 14
o
17
18
9) What is the graph of 7 ≤ m?
1.
2.
3.
4.
6
6
6
6
o
7
o
7
●
7
●
7
Answer Now
8
8
8
8
10) Solve
1.
2.
3.
4.
5.
6.
7.
Draw “the river”
Subtract 2r from both
sides
Simplify
Add 17 to both sides
Simplify
Check your answer
Graph the solution
30
3r - 17 ≥ 2r + 14
- 2r
- 2r
r – 17 ≥ 14
+ 17 + 17
r ≥ 31
3(31) - 17 = 2(31) + 14
●
31
32
11) Solve -2x + 6 ≥ 3x - 4
1.
2.
3.
4.
x≥
x≤
x≥
x≤
-2
-2
2
2
Answer Now
12) Joanna’s tests were 87, 93, 88
and 94. What must her 5th grade be
to get a total of at least 459?
1.
2.
3.
4.
96
97
98
100
Answer Now
Objective
The student will be able to:
solve inequalities using multiplication and
division.
Designed by Skip Tyler, Varina High School
1) Which graph represents the
k
correct answer to
>1
4
1.
2.
3.
4.
-5
-5
-5
-5
o
-4
o
-4
●
-4
●
-4
Answer Now
-3
-3
-3
-3
1) Solve
1.
2.
3.
4.
5.
6.
Draw “the river”
Clear the fraction Multiply both sides by -4
NEW STEP!!
When multiplying BOTH
sides by a NEGATIVE
number, SWITCH the
inequality!
Simplify
Check your answer
Graph the solution
-53
4
k
>
13
4
k

13 -4
4
k < -52
52
 13
4
o
-52
-51
x
>
-10
3
2) When solving
will the inequality switch?
1. Yes!
2. No!
3. I still don’t
know!
Answer Now
1.
2.
x
2) Solve
< -10
3
Draw “the river”
x
Clear the fraction 3
 10 3
Multiply both sides by 3.
3
Do you switch the
inequality?
x < -30
30
 10
3
No - Both sides are being
multiplied by a positive
number
3.
4.
5.
Simplify
Check your answer
Graph the solution
-31
o
-30
-29
a
6
3) When solving
4
will the inequality switch?
1. Yes!
2. No!
3. I still don’t
know!
Answer Now
1.
2.
a
6
3) Solve
4
Draw “the river”
a
Clear the fraction -4
 6 -4
Multiply both sides by -4.
4
Do you switch the
inequality?
a > -24
Yes - Both sides are being
multiplied by a negative
number
3.
4.
5.
(24)
6
4
Simplify
Check your answer
Graph the solution
-25
o
-24
-23
Take a quick break!
Things that make you go.....Hmmm
Why is abbreviation such a long
word?
Why do they have braille dots on the
ATM machines in drive-thru’s?
4) Solve -8p ≥ -96
1.
2.
3.
4.
p ≥ 12
p ≥ -12
p ≤ 12
p ≤ -12
Answer Now
-8p ≥ -96
8 p 96

8
8
p ≤ 12
-8(12) = -96
4) Solve
1.
2.
Draw “the river”
Divide both sides by -8.
Do you switch the
inequality?
Yes - Both sides are being divided
by a negative number
3.
4.
5.
Simplify
Check your answer
Graph the solution
11
●
12
13
5) Solve 7v < -105
1.
2.
3.
4.
o
-16 -15 -14
o
-16 -15 -14
●
-16 -15
●
-14
-15 -15 -14
Answer Now
Objective
The student will be able to:
solve two-step inequalities.
SOL: A.5abc
Designed by Skip Tyler, Varina High School
1.
2.
3.
4.
5.
6.
7.
1) Solve 5m - 8 > 12
Draw “the river”
+8 +8
Add 8 to both sides
5m
> 20
Simplify
Divide both sides by 5
5
5
Simplify
Check your answer
m>4
Graph the solution
5(4) – 8 = 12
3
o
4
5
1.
2.
3.
4.
5.
2) Solve 12 - 3a > 18
Draw “the river”
- 12
- 12
Subtract 12 from both
-3a > 6
sides
Simplify
-3 -3
Divide both sides by -3
Simplify
a
<
-2
(Switch the inequality!)
Check your answer
12 - 3(-2) = 18
6.
7. Graph the solution
-3
o
-2
-1
Which graph shows the solution to
2x - 10 ≥ 4?
1.
.
2.
3.
4.
Answer Now
3) Solve 5m - 4 < 2m + 11
1. Draw “the river”
2. Subtract 2m from both
sides
3. Simplify
4. Add 4 to both sides
5. Simplify
6. Divide both sides by 3
7. Simplify
8. Check your answer
9. Graph the solution
4
-2m
-2m
3m - 4 < 11
+4 +4
3m < 15
3
3
m<5
5(5) – 4 = 2(5) + 11
o
5
6
4) Solve 2r - 18 ≤ 5r + 3
1.
2.
3.
4.
5.
6.
7.
8.
9.
Draw “the river”
Subtract 2r from both sides
Simplify
Subtract 3 from both sides
Simplify
Divide both sides by 3
Simplify
Check your answer
Graph the solution
-8
-2r
-2r
-18 ≤ 3r + 3
-3
-3
-21 ≤ 3r
3
3
-7 ≤ r or r ≥ -7
2(-7) – 18 = 5(-7) + 3
●
-7
-6
6) Solve -2x + 6 ≥ 3x - 4
1.
2.
3.
4.
x≥
x≤
x≥
x≤
-2
-2
2
2
Answer Now
5) Solve
1.
2.
3.
4.
5.
6.
7.
8.
9.
Draw “the river”
Subtract 14p from both
sides
Simplify
Add 20 to both sides
Simplify
Divide both sides by 12
Simplify
Check your answer
Graph the solution
6
26p - 20 > 14p + 64
-14p
-14p
12p – 20 > 64
+ 20 + 20
12p > 84
12
12
p>7
26(7) – 20 = 14(7) + 64
o
7
8
What are the values of x if
3(x + 4) - 5(x - 1) < 5?
1.
2.
3.
4.
x < -6
x > -6
x<6
x>6
Answer Now
Objectives
The student will be able to:
1. solve compound inequalities.
2. graph the solution sets of
compound inequalities.
Designed by Skip Tyler, Varina High School
What is the difference between
and and or?
A
B
A
B
AND means intersection
-what do the two items
have in common?
OR means union
-if it is in one item, it is in
the solution
1) Graph x < 4 and x ≥ 2
a) Graph x < 4
o
2
3
4
3
4
b) Graph x ≥ 2
●
2
c) Combine the graphs
d) Where do they intersect?
●
2
3
o
4
2) Graph x < 2 or x ≥ 4
a) Graph x < 2
o
2
b) Graph x ≥ 4
2
3
3
c) Combine the graphs
4
●
4
3) Which inequalities describe the
following graph?
o
o
1.
2.
3.
4.
-3
-2
y > -3 or y < -1
y > -3 and y < -1
y ≤ -3 or y ≥ -1
y ≥ -3 and y ≤ -1
Answer Now
-1
4) Graph the compound inequality
6<m<8
When written this way, it is the same thing as
6 < m AND m < 8
It can be rewritten as m > 6 and m < 8 and
graphed as previously shown, however,
it is easier to graph everything
between 6 and 8!
o
6
7
o
8
5) Which is equivalent to
-3 < y < 5?
1.
2.
3.
4.
y > -3 or y < 5
y > -3 and y < 5
y < -3 or y > 5
y < -3 and y > 5
Answer Now
1.
2.
3.
4.
6) Which is equivalent to
x > -5 and x ≤ 1?
-5 < x ≤ 1
-5 > x ≥ 1
-5 > x ≤ 1
-5 < x ≥ 1
Answer Now
7) 2x < -6 and 3x ≥ 12
1. Solve each inequality
for x
2. Graph each inequality
3. Combine the graphs
4. Where do they
intersect?
5. They do not! x cannot
be greater than or equal
to 4 and less than -3
No Solution!!
2 x 6

2
2
x  3
o
-3
3x 12

3
3
x4
0
4
7
-6
1
o
●
8) Graph 3 < 2m – 1 < 9
Remember, when written like this, it is an
AND problem!
3 < 2m – 1 AND 2m – 1 < 9
Solve each inequality.
Graph the intersection of
2 < m and m < 5.
-5
0
5
9) Graph x < 2 or x ≥ 4
-5
0
5
10) Graph x ≥ -1 or x ≤ 3
-5
0
5
The whole line is shaded!!