Transcript alg 1 ch 6.1 & 2 2012

Algebra 1 ~ Chapter 6.1 and
6.2
Solving One-Step Inequalities
 Recall that statements with greater than
(>), less than (<), greater than or equal to (≥),
or less than or equal to (≤) are inequalities
 Solving one-step inequalities is much like
solving one-step equations.
 To solve an inequality, you need to isolate
the variable using the properties of inequality
and inverse operations.
Ex. 1 - Solve the inequality and graph the solutions.
x + 12 < 20
–12 –12
x <
–10 –8 –6 –4 –2
The solution set is {all numbers
8
0
less than 8}.
2
4
6
8 10
The heavy arrow pointing to the
left shows that the inequality
includes all #s less 8.
The circle at 8 is open.
This shows that 8 is NOT
included in the inequality.
Check your solution??
Ex. 2 - Solve the inequality and graph the solutions.
d – 5 > –7
+5
Example check:
+5
d=0
d > –2
d – 5 > -7
0 – 5 > -7
-5 > -7
–10 –8 –6 –4 –2
0
2
4
6
8 10
TRUE!
Example check:
d = -6
d – 5 > -7
-6 - 5 > -7
-11 > -7
FALSE!
Ex. 3 – Solving an Inequality with Variables on both sides
Solve
12x – 4 ≤ 13x
-12x
-12x
-4 ≤ 1x
x ≥ -4
Ex. 4 - Solve the inequality and graph the solutions.
CHECK
7x > –42
7x > -42
7(0) > -42
x > –6
0 > -42
TRUE!
–10 –8 –6 –4 –2
0
2
4
6
8 10
CHECK
7x > -42
7(-10)>-42
-70 > -42
FALSE!
Ex. 5 - Solve the inequality and graph the solutions.
m
2
3
3(2) ≤ 3
6 ≤ m (or m ≥ 6)
0
2
4
6
8 10 12 14 16 18 20
Check:
m
2
3
12
2
3
24
Ex. 6 - Solve the inequality and graph the solutions.
Since r is multiplied by ,
multiply both sides by the
reciprocal of
r < 16
0
2
4
6
8 10 12 14 16 18 20
.
What happens when you multiply or divide both sides
of an inequality by a negative number?
Look at the number line below.
–b
a<b
–a > –b
–a
0
Multiply both
sides by –1.
You can tell from the
number line that
–a > –b.
a
b
b > –a
–b < a
Multiply both
sides by –1.
You can tell from the
number line that
–b < a.
Notice that when you multiply (or divide) both sides of an
inequality by a negative number, you must reverse the
inequality symbol.
A real number example…
3 < 8
-1(3)
-3
?
Start off with the # 8 is
greater than the # 3. TRUE!
-1(8)
> -8
Flip it!!
If I multiply both sides by a
negative # (-1 in this case)…
In order to keep this inequality
TRUE, what must I do to the
inequality symbol?
Caution!
Do not change the direction of the inequality
symbol just because you see a negative
sign. For example, you do not change the
symbol when solving 4x < –24.
Ex. 7 - Solve the inequality and graph the solutions.
–12x > 84
Since x is multiplied by –12, divide
both sides by –12. Change > to <.
x < –7
CHECK
–7
-12x > 84
–14 –12 –10 –8 –6 –4 –2
0
2
4
6
-12(-12) > 84
144 > 84
TRUE!
Ex. 8 - Solve the inequality and graph the solutions.
Since x is divided by –3, multiply
both sides by –3. Change to .
24  x (or x  24)
10 12 14 16 18 20 22 24 26 28 30
Ex. 9 - Solve the inequality and graph the
solutions. Check your answer.
10 ≥ –x
–1(10) ≤ –1(–x)
Multiply both sides by –1 to make x
positive. Change  to .
–10 ≤ x (or even better x ≥ -10)
CHECK
10 ≥ –x
10 ≥ -(4)
–10 –8 –6 –4 –2
0
2
4
6
8 10
10 ≥ -4
TRUE!
Ex. 10 – Define a variable and write an inequality for
each problem. You do not need to solve the inequality.
a.) A number decreased by 8 is at most 14.
n – 8 ≤ 14
b.) A number plus 7 is greater than 2.
n+7>2
c.) Half of a number is at least 26.
½n ≥ 26
Lesson Wrap Up
Solve each inequality and graph the solutions.
1. 13 < x + 7
x>6
2. –6 + h ≥ 15
h ≥ 21
3.
x > 20
4. –5x ≥ 30
x ≤ –6