Download 2.6

Goals:
Solving Inequalities
1. Use graphs to represent the solution
set of an inequality.
2. Use set-builder notation to represent
solution set of an inequality.
3. Apply addition and multiplication
principles to solving inequalities.
A solution to an inequality is a value of the
variable that makes the inequality true.
Inequalities relate variables and
numbers with greater than (>),
greater than or equal to (≥), less
than (<), or less than or equal to
(≤) symbols.
Examples:
x = -5, x = 2, x = 5 are all solutions to the
inequality
5 ≥ x,
but x = 5 is not a solution to the inequality
5 > x.
Number lines are often used to represent
all solutions to an inequality.
Represents the solutions to
5 ≥ x.
The filled-in point at 5 means include 5 in
the solution set.
This represents the solutions to x < 5.
The open circle at x = 5 indicates that x =
5 is not in the solution set.
Open circle – exclude that point.
Closed circle – include that point.
1
Addition Principle for Inequalities:
For any real numbers a, b, and c;
Equivalent inequalities: Two
inequalities with the same
solution set are said to be
equivalent.
a < b is equivalent to a + c < b + c
a ≤ b is equivalent to a + c ≤ b + c
a > b is equivalent to a + c > b + c
a ≥ b is equivalent to a + c ≥ b + c
Same as with equalities.
Example:
x + 7 > 12 is equivalent to x > 5 because
x + 7 > 12
−7 −7
x > 5
Multiplication Principle for Inequalities:
For any real numbers a and b, and for any
positive number c:
a < b is equivalent to ac < bc and
a > b is equivalent to ac > bc.
But, if c is negative
a < b is equivalent to ac > bc and
a > b is equivalent to ac < bc.
Multiplying by a negative flips the
inequality. The same is true for ≥ and ≤.
Why does multiplying by a negative flip
the inequality?
Examples:
1. 2x < 6
We know that 5 < 7. The multiplication
principal tells us that
.
5 2<
7.
2 or 10 < 14.
2. −
1
x≥9
3
But is we multiply both side by –2, we get
–10 > –14;
Only true because we flipped the
inequality.
3. – 5x + 2 > –8
Check!
2
Set-builder notation. The set x < 2 can
be written
{x | x < 2}
which is read, “the set of all x such that
x is less than 2.”
3