Section 1.5: Solving Inequalities
Objectives:
To solve and graph inequalities.
To write and solve compound inequalities.
 Words like “at most” and “ at least” suggest a relationship in which two quantities may not be equal.
Inequality
Word Sentence
x> 4
x is greater than 4.
x≥4
x is greater than or equal to 4.
x is at least 4.
x< 4
x is less than 4.
x≤4
x is less than or equal to 4.
x is at most 4.
Verbal to Algebraic
EX: 5 fewer than a number is at least 12

“fewer” indicates subtraction

“at least” indicates greater than or equal to
x – 5 ≥ 12
EX: Solve the inequality -3 (2x – 5) + 1 ≥ 4
-3 (2x – 5) + 1 ≥ 4
-6x + 15 + 1 ≥ 4
-6x + 16 ≥ 4
NOTE:
-6x ≥ -12
x≤2
Sign flipped when dividing
by a negative number.
Set Builder Notation:
{x| x>9}
Interval Notation:
(- ∞, 2) or ( 2, ∞)
Symbols:
When using ≥ or ≤ use brackets [ ] and closed circles
When using > or < use parentheses ( ) and open circles
Graph
4
4
4
4
Graphing Inequalities
EX: - m ≤
m4
9
-9m ≤ m + 4
-10m ≤ 4
m≥
NOTE:
2
5
-1
Sign flipped when dividing
by a negative number.
0
1
2
Steps for Graphing an Inequality

Once you solve for the variable, make a number line including that number.

If the sign is < or > then indicate the solution with an open circle.

If the sign is ≤ or ≥ then indicate the solution with a closed circle.

Shade the number line in the direction of the symbol.
NO Solution or All Real Numbers
EX: -2 (3x + 1) > -6x + 7
-6x – 2 > -6x + 7
+6x
+6x
-2 > 7
Since -2 is NEVER greater than 7, the inequality is always false. Therefore, the answer has no solution.
EX: 5(2x – 3) – 7x ≤ 3x + 8
10x – 15 – 7x ≤ 3x + 8
3x – 15 ≤ 3x + 8
-3x
-3x
-15 ≤ 8
The inequality -15 ≤ 8 is true, so the inequality is always true. Therefore, all real numbers are solutions.
AND Inequality
OR Inequality