Chapter 1
Section 4
Solving Inequalities
Inequalities

The solutions include more than one number
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All of the rules for solving equations apply to
inequalities, with one added:
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
Ex: 2 < x ;values that x could be include 3, 7, 45…
If you multiply or divide by a NEGATIVE you must FLIP the
sign. (< becomes > and > becomes <)
When graphing on a number line:



Open dot for < or >
Closed (solid) dot for ≤ or ≥
The shading should be easy to see (a slightly elevated line is
ok) --- see examples
Solving Inequalities
ALGEBRA 2 LESSON 1-4
Solve –2x < 3(x – 5). Graph the solution.
–2x < 3(x – 5)
–2x < 3x – 15 Distributive Property
–5x < –15
x >3
Subtract 3x from both sides.
Divide each side by –5 and reverse
the inequality.
Inequalities – Special Solution
Cases


If the variables cancel, and you’re left
with a true statement (ex. 0<10) then
all numbers are solutions for the
inequality.
If the variables cancel, and you’re left
with a false statement (ex. 0>10) then
no numbers are solutions for the
inequality.
Solve 7x ≥ 7(2 + x). Graph the solution.
7x ≥ 7(2 + x)
7x ≥ 14 + 7x
0 ≥ 14
Distributive Property
Subtract 7x from both sides.
0
The last inequality is always false, so 7x ≥ 7(2 + x) is
always false. It has no solution.
Compound Inequalities Disjuctions



Compound Inequality – a pair of
inequalities joined by or
For or statements the value must satisfy
one of the inequalities
Example:
x < -1 or x ≥ 3
Or Inequalities
ALGEBRA 2 LESSON 1-4
Graph the solution of 3x + 9 < –3 or –2x + 1 < 5.
3x + 9 < –3
or
–2x + 1 < 5
3x < –12
or
-2x < 4
x < –4 or x > –2
Try This Problem
Graph the solution of x – 1 < 3 or x - 3 > 8
x < 4 or x > 11
Compound Inequalities Conjuctions



A pair of inequalities joined by an and
statement
The value must satisfy both inequalities.
In other words, the solution is where
the inequalities overlap.
Can be written in two forms:
Example: -1 < x and x ≤ 3 is the same
as -1 < x ≤ 3
And Inequalities
a)
Graph the solution of
3x – 1 > -28 and 2x + 7 < 19.
3x > -27 and 2x < 12
x > -9 and x < 6
b)
Graph the solution of -8 < 3x + 1 <19
-9 < 3x < 18
-3 < x < 6
Check for Understanding
Graph the solution of 2x > x + 6 and x – 7 < 2
a)
a)
x > 6 and x < 9