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1.6 Solve Linear Inequalities
Goal  Solve linear inequalities.
Your Notes
VOCABULARY
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  0: ax + b< 0, ax + b > 0, ax + b  0, ax + b  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.
Checkpoint Graph the inequality.
1. x  1
2. x > 3
Your Notes
Example 2
Graph compound inequalities
a. Graph 3 < x < 1.
b Graph x < 1 or x  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_.
Checkpoint 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  1
x  _4_
Multiply each side by the
same positive number.
1
x > 10
2
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  45
x _ 5_
Your Notes
Example 3
Solve an inequality with a variable on both sides
Solve 4x + 5 > 9x  10. Then graph the solution.
4x + 5 > 9x  10
_5x_+ 5 > 10
5x >_15_
x < __3___
Write original inequality.
Subtract _9x_ from each side.
Subtract _5_ from each side.
Divide each side by_5_ and
_reverse_ the inequality.
The solutions are real numbers _less_ than _3_.
Checkpoint Solve the inequality. Then graph the solution.
4. x  2  4x  8
x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
7 + _2_ < 5x 2 + _2_  8 _2_
_5_ < 5x  _10_
_1_ < x  _2_
Write original inequality.
Add 2 to each expression.
Simplify.
Divide each expression by _5_.
The solutions are real numbers _greater_ than _1_ and _less_ than or equal to_2_
Your Notes
Example 5
Solve an "or" compound inequality
Solve 4x  7  5 or 3x + 2  23. Then graph the solution.
Solution
The solution of this inequality is a solution of either of its parts.
4x  7  5
First Inequality
Write inequality
4x  _12_
x  _3_
Add _7_ to each side.
Divide each side by_4_.
Second inequality.
3x + 2  23
Write inequality
3x  _21_
x  _7_
Subtract _2_ from
each side
Divide each side by
_3_.
The solutions are all real numbers _less_ than or equal to _3_ or _greater_ than or equal
to _7_
Checkpoint 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  26
x < 4 or x  5
Homework
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