Download Lesson 2

Inequalities with
2-5 Solving
Variables on Both Sides
What you need…….MAKE SURE YOU HAVE
TURNED IN:
Riddle WS
Any work from last week… Thursday is last day to
turn in work!!!
TEST NEXT MONDAY! ONE PAGE OF NOTES ONLY
PHONES AWAY
New Warm-Up Sheet!
Pencil/Pen
Notes
Holt McDougal Algebra 1
Inequalities with
2-5 Solving
Variables on Both Sides
Warm Up 10/31
Solve each equation.
1. 2x = 7x + 15 x = –3
2. 3y – 21 = 4 – 2y
y=5
3. 2(3z + 1) = –2(z + 3) z = –1
4. 3(p – 1) = 3p + 2
Holt McDougal Algebra 1
no solution
Inequalities with
2-5 Solving
Variables on Both Sides
Objective
Solve inequalities that contain variable
terms on both sides.
Holt McDougal Algebra 1
Inequalities with
2-5 Solving
Variables on Both Sides
Some inequalities have variable terms on both
sides of the inequality symbol. You can solve
these inequalities like you solved equations with
variables on both sides.
Use the properties of inequality to “collect” all the
variable terms on one side and all the constant
terms on the other side.
Holt McDougal Algebra 1
Inequalities with
2-5 Solving
Variables on Both Sides
Example 1A: Solving Inequalities with Variables on
Both Sides
Solve the inequality and graph the solutions.
y ≤ 4y + 18
To collect the variable terms on one
y ≤ 4y + 18
side, subtract y from both sides.
–y –y
0 ≤ 3y + 18
–18
– 18
Since 18 is added to 3y, subtract 18
from both sides to undo the
addition.
–18 ≤ 3y
Since y is multiplied by 3, divide both
sides by 3 to undo the
multiplication.
–6 ≤ y (or y  –6)
Holt McDougal Algebra 1
–10 –8 –6 –4 –2
0
2
4
6
8 10
Inequalities with
2-5 Solving
Variables on Both Sides
Example 1B: Solving Inequalities with Variables on
Both Sides
Solve the inequality and graph the solutions.
4m – 3 < 2m + 6
To collect the variable terms on one
–2m
– 2m
side, subtract 2m from both sides.
2m – 3 <
+6
+3
+3
2m
<
9
Since 3 is subtracted from 2m, add
3 to both sides to undo the
subtraction
Since m is multiplied by 2, divide
both sides by 2 to undo the
multiplication.
4
Holt McDougal Algebra 1
5
6
Inequalities with
2-5 Solving
Variables on Both Sides
Go get a white board, marker and eraser
Holt McDougal Algebra 1
Inequalities with
2-5 Solving
Variables on Both Sides
Check It Out! Example 1a
Solve the inequality and graph the solutions.
4x ≥ 7x + 6
4x ≥ 7x + 6
–7x –7x
To collect the variable terms on one
side, subtract 7x from both sides.
–3x ≥ 6
x ≤ –2
Since x is multiplied by –3, divide
both sides by –3 to undo the
multiplication. Change ≥ to ≤.
–10 –8 –6 –4 –2
Holt McDougal Algebra 1
0
2
4
6
8 10
Inequalities with
2-5 Solving
Variables on Both Sides
Check It Out! Example 1b
Solve the inequality and graph the solutions.
5t + 1 < –2t – 6
5t + 1 < –2t – 6
+2t
+2t
7t + 1 < –6
– 1 < –1
7t
< –7
7t < –7
7
7
t < –1
–5 –4 –3 –2 –1
Holt McDougal Algebra 1
0
1
2
To collect the variable terms on
one side, add 2t to both sides.
Since 1 is added to 7t, subtract 1
from both sides to undo the
addition.
Since t is multiplied by 7, divide
both sides by 7 to undo the
multiplication.
3
4
5
Inequalities with
2-5 Solving
Variables on Both Sides
You may need to simplify one or both sides of
an inequality before solving it. Look for like
terms to combine and places to use the
Distributive Property.
Holt McDougal Algebra 1
Inequalities with
2-5 Solving
Variables on Both Sides
What you need…….MAKE SURE YOU HAVE
TURNED IN:
Riddle WS
Any work from last week… Thursday is last day to
turn in work!!!
TEST NEXT MONDAY! ONE PAGE OF NOTES ONLY
PHONES AWAY
Warm-Up Sheet!
Pencil/Pen
Holt McDougal Algebra 1
Inequalities with
2-5 Solving
Variables on Both Sides
Warm-Up 11/1
Solve the inequality and graph the solutions.
5(2 – r) ≥ 3(r – 2)
Distribute 5 on the left side of the
inequality and distribute 3 on
5(2 – r) ≥ 3(r – 2)
the right side of the inequality.
5(2) – 5(r) ≥ 3(r) + 3(–2)
Since 6 is subtracted from 3r,
10 – 5r ≥ 3r – 6
add 6 to both sides to undo
+6
+6
the subtraction.
16 − 5r ≥ 3r
Since 5r is subtracted from 16
+ 5r +5r
add 5r to both sides to undo
the subtraction.
16
≥ 8r
Holt McDougal Algebra 1
Inequalities with
2-5 Solving
Variables on Both Sides
Check It Out! Example 3a Continued
16 ≥ 8r
Since r is multiplied by 8, divide
both sides by 8 to undo the
multiplication.
2≥r
–6
–4
Holt McDougal Algebra 1
–2
0
2
4
Inequalities with
2-5 Solving
Variables on Both Sides
Example 3A: Simplify Each Side Before Solving
Solve the inequality and graph the solutions.
2(k – 3) > 6 + 3k – 3
Distribute 2 on the left side of
2(k – 3) > 3 + 3k
the inequality.
2k + 2(–3) > 3 + 3k
2k – 6 > 3 + 3k
–2k
– 2k
–6 > 3 + k
–3 –3
–9 > k
Holt McDougal Algebra 1
To collect the variable terms,
subtract 2k from both
sides.
Since 3 is added to k, subtract 3
from both sides to undo the
addition.
Inequalities with
2-5 Solving
Variables on Both Sides
Example 3A Continued
–9 > k
–12
–9
Holt McDougal Algebra 1
–6
–3
0
3
Inequalities with
2-5 Solving
Variables on Both Sides
Example 3B: Simplify Each Side Before Solving
Solve the inequality and graph the solution.
0.9y ≥ 0.4y – 0.5
0.9y ≥ 0.4y – 0.5
–0.4y –0.4y
To collect the variable terms,
subtract 0.4y from both sides.
0.5y ≥
– 0.5
0.5y ≥ –0.5
0.5
0.5
y ≥ –1
–5 –4 –3 –2 –1
Holt McDougal Algebra 1
0
1
2
3
Since y is multiplied by 0.5,
divide both sides by 0.5 to
undo the multiplication.
4
5
Inequalities with
2-5 Solving
Variables on Both Sides
Check It Out! Example 3b
Solve the inequality and graph the solutions.
0.5x – 0.3 + 1.9x < 0.3x + 6
Simplify.
2.4x – 0.3 < 0.3x + 6
Since 0.3 is subtracted
2.4x – 0.3 < 0.3x + 6
from 2.4x, add 0.3 to
+ 0.3
+ 0.3
both sides.
2.4x
< 0.3x + 6.3
Since 0.3x is added to
–0.3x
–0.3x
6.3, subtract 0.3x from
both sides.
2.1x
<
6.3
Since x is multiplied by
2.1, divide both sides
by 2.1.
x<3
Holt McDougal Algebra 1
Inequalities with
2-5 Solving
Variables on Both Sides
Check It Out! Example 3b Continued
x<3
–5 –4 –3 –2 –1
Holt McDougal Algebra 1
0
1
2
3
4
5