1.6 Solve Linear Inequalities
Linear
inequality
A linear inequality in one variable can be written
in one of the following forms, where a and b are
real numbers and a ¹ 0: ax + b<
0, ax + b > 0, ax + b £ 0, ax + b ³ 0.
Compound
inequality
Consists
of two simple inequalities joined by "and" or "or"
Equivalent
inequalities
Inequalities
that have the same solutions as the original inequality
Example
1
Graph
simple inequalities
a.
Graph x £ 4. b.
Graph x > -2.
Solution
a.
The
solutions are all real numbers _less than_ or _equal to_4. A _closed_
dot is used in the graph to indicate 4 is a solution.
b. The solutions are all real numbers _greater
than_-2. An _open_ dot is used in the graph
to indicate -2 is not a solution.
Graph the inequality.
1.
x £ -1
2.
x > -3
Example
2
Graph
compound inequalities
a.
Graph -3 < x < 1. b Graph x < -1 or x ³ 1.
Solution
a. The solutions are all real numbers that
are _greater_ than _-3_ and _less_
than _1_.
b. The solutions are all real numbers that
are _less_ than _-1_ or _greater
than_ or _equal_ to_1_.
Graph
the inequality.
3.
x £ 0 or x > 2
TRANSFORMATIONS THAT PRODUCE EQUIVALENT INEQUALITIES
|
Transformation Applied
to Inequality |
Original Inequality |
Equivalent Inequality |
|
Add the same number to each
side. |
x - 7 < 4 |
x < _11_ |
|
Subtract the same number
from each side. |
x + 3 ³ -1 |
x ³ _-4_ |
|
Multiply each side by the
same positive number. |
|
x > _20_ |
|
Divide each side by the
same positive number. |
5x £ 15 |
x > _3_ |
|
Multiply each side by the
same negative number and reverse the inequality. |
-x < 17 |
x _> - 17_ |
|
Divide each side by the
same negative number and reverse the inequality. |
-9x ³ 45 |
x _£ -5_ |
Example
3
Solve
an inequality with a variable on both sides
Solve
4x + 5 > 9x - 10. Then graph the solution.
|
4x + 5 > 9x
- 10 |
Write original inequality. |
|
_-5x_+ 5 > -10 |
Subtract _9x_
from each side. |
|
-5x >_-15_ |
Subtract _5_ from
each side. |
|
x < __3___ |
Divide each side by_-5_ and _reverse_
the inequality. |
The
solutions are real numbers _less_ than _3_.
Solve
the inequality. Then graph the solution.
4.
x - 2 £ 4x - 8
x ³ 2
5.
-7x + 6 > -1
x
< 1
Example
4
Solve
an "and" compound inequality
Solve
-7 < 5x - 2 £ 8. Then graph the solution.
|
-7 < 5x - 2 £ 8 |
Write original inequality. |
|
-7 + _2_ < 5x -2 + _2_ £ 8 _2_ |
Add 2 to each expression. |
|
_-5_ <
5x £ _10_ |
Simplify. |
|
_-1_ < x £ _2_ |
Divide each expression by _5_. |
The
solutions are real numbers _greater_ than _-1_ and _less_
than or equal to_2_
Example
5
Solve an "or"
compound inequality
Solve
4x - 7 £ 5 or 3x + 2 ³ 23. Then graph the
solution.
Solution
The solution of this
inequality is a solution of either of its parts.
|
First Inequality |
Second inequality. |
|
4x - 7 £ 5 Write inequality |
3x + 2 ³ 23 Write inequality |
|
4x £ _12_ Add
_7_ to each side. |
3x ³ _21_ Subtract _2_ from each side |
|
x £ _3_ Divide
each side by_4_. |
x ³ _7_ Divide each side by _3_. |
The solutions are all real
numbers _less_ than or equal to _3_ or _greater_ than or
equal to _7_
Solve the inequality. Then graph the solution.
6.
-3 < 4x + 5 < 21
-2 < x < 4
7.
-9 < 2x - 3 £ 1
-3 < x £ 2
8.
2x - 6 < -2 or 5x + 1 ³ 26
x < 4 or x ³ 5