Pre-CalculusLesson 7: Solving InequalitiesLinear inequalities, compound inequalities, absolute value inequalities, interval notation
Number Line
< and > are shown with open circles
x<2 x>4
Number Line
< and > are shown with open circles
x<2 x>4
Number Line
< and > are shown with open circles
x<2 x>4
Number Line
and are shown with closed circles
x 2 x 4
Number Line
and are shown with closed circles
x 2 x 4
Number Line
and are shown with closed circles
x 2 x 4
Multiplication Property of Inequality
When multiplying or dividing by a negative number, FLIP the INEQUALITY SIGN!
Example:
756 y66 15 y
Example:
756 y66 15 y
55
Example:
756 y66 15 y
55
5
1y
Compound Inequalities
Conjunction Example #17432 x
-3 -2 -1 0 1 2
Conjunction Example #17432 x
444
-3 -2 -1 0 1 2
Conjunction Example #17432 x
444 336 x
-3 -2 -1 0 1 2
Conjunction Example #17432 x
444 336 x333
12 x
-3 -2 -1 0 1 2
Conjunction Example#2634 x
6 7 8 9 10 11
Conjunction Example#2634 x333
97 x
6 7 8 9 10 11
Conjunction Example#2634 x333
97 x111
97 x
6 7 8 9 10 11
Disjunction Example#13434 xorx
0 1 2 3 4 5 6 7 8 9 10
Disjunction Example#13434 xorx4444
71 xorx
0 1 2 3 4 5 6 7 8 9 10
Disjunction Example#235342 xorx
-5 -4 -3 -2 -1 0 1 2 3 4 5
Disjunction Example#235342 xorx
5544 272 xorx
-5 -4 -3 -2 -1 0 1 2 3 4 5
Disjunction Example#235342 xorx
5544 272 xorx
-5 -4 -3 -2 -1 0 1 2 3 4 5
22
22
7 xorx
Absolute Value Inequalities
“Less Than”
Rewrite the inequality as a conjunction.
-a < x < a
Solve.
-4 -3 -2 -1 0 1 2
Example 743 x
-4 -3 -2 -1 0 1 2
Example
7437 x
743 x
-4 -3 -2 -1 0 1 2
Example
7437 x444
3311 x
743 x
-4 -3 -2 -1 0 1 2
Example
7437 x444
3311 x333
13
11 x
743 x
“Greater Than”
Rewrite the inequality as a disjunction.
x < -a or x > a
Solve.
Example
-5 -4 -3 -2 -1 0 1 2 3 4 5
342 x
Example342342 xorx
-5 -4 -3 -2 -1 0 1 2 3 4 5
342 x
Example342342 xorx
4444 1272 xorx
-5 -4 -3 -2 -1 0 1 2 3 4 5
342 x
Example342342 xorx
4444 1272 xorx
-5 -4 -3 -2 -1 0 1 2 3 4 5
2222
2
1
2
7 xorx
342 x
Interval Notation When using interval notation:
( means "not included" or "open". [ means "included" or "closed".
The inequality would be written as the interval The inequality would be written as the interval
62 x 6,232 xorx
,32,
Which statement below is the correct interval notation for the situation depicted in this number line graph?
http://www.regentsprep.org/regents/math/ALGEBRA/AP1/IntPrac.htm
Which statement below is the correct interval notation for the situation depicted in this number line graph?
http://www.regentsprep.org/regents/math/ALGEBRA/AP1/IntPrac.htm
Write the following statement as an inequality:
x < -3 or 0 < x < 2 or x > 4 x < -3 or 0 < x < 2 or x > 4 x < -3 or 0 < x < 2 or x > 4
http://www.regentsprep.org/regents/math/ALGEBRA/AP1/IntPrac.htm
Write the following statement as an inequality:
x < -3 or 0 < x < 2 or x > 4 x < -3 or 0 < x < 2 or x > 4 x < -3 or 0 < x < 2 or x > 4
http://www.regentsprep.org/regents/math/ALGEBRA/AP1/IntPrac.htm
Write the following inequality as interval notation: -2 < x < 1 or x > 1
http://www.regentsprep.org/regents/math/ALGEBRA/AP1/IntPrac.htm
Write the following inequality as interval notation: -2 < x < 1 or x > 1
http://www.regentsprep.org/regents/math/ALGEBRA/AP1/IntPrac.htm
Practice QuestionsSolve each inequality, express the answer in interval notation, and graph the solution on the number line.1.
2.
3.
4.
5.