Algebra II
1-4: Solving Inequalities Day 2
A compound inequality is a pair of inequalities joined by and (conjunction) or or (disjunction).
For a conjunction (and), you can sometimes use a special notation called a betweenness.
For example, x > 2 and x < 4 could also be expressed as 2 < x < 4.
You cannot do this with a disjunction. A common error is to attempt this. For example, some
people will try to combine x > 3 or x < 1 as 1 > x > 3, but this is incorrect, because this
statement indicates that 1 is greater than 3, and that is clearly false.
You should also be careful not to make statements such as 1 < x > 3 or 1 > x < 3. If you are
using two inequality symbols in the same expression, they must point in the same direction.
Ex. 1:
Solve the compound inequality. Graph the solution.
6x ≥ −24 and 9x < 54
x ≥ −4 and x < 6, which can be written as −4 ≤ x < 6 .
-6
-4
-2
0
2
4
6
8
10
12
Ex. 2:
Solve the compound inequality. Graph the solution.
4x < 16 or 12x > 144
x < 4 or x > 12
3
4
5
6
7
8
9
10
11
12
Ex. 3:
Solve the compound inequality. Graph the solution.
6c ≤ 18 or −5c ≤ 15
c ≤ 3 or c ≥ −3 (Remember to reverse the inequality when you divide by a negative number.)
All real numbers are solutions.
Ex. 4:
By how much should a machinist decrease the length of a rod that is 4.78 cm long if the length
must be within 0.02 cm of 4.5 cm?
4.5 − 0.02 ≤ 4.78 − x ≤ 4.5 + 0.02
4.48 ≤ 4.78 − x ≤ 4.52
4.48 − 4.78 ≤ −x ≤ 4.52 − 4.78
−0.30 ≤ −x ≤ −0.26
0.30 ≥ x ≥ 0.26
The machinist should remove between 0.26 cm and 0.30 cm.