Learning Objectives
Read strict and inclusive inequality symbols.
Solve one- and two-step linear inequalities.
Describe a solution set on a number line and check representative values.
Prerequisites: comparing signed numbers and solving equations. An inequality compares sizes instead of asserting equality. Its solution is often an entire set of numbers.
1. Read the Boundary Carefully
Symbol | Meaning | Boundary included? |
|---|---|---|
x < 5 | x is less than 5 | No |
x > 5 | x is greater than 5 | No |
x ≤ 5 | x is at most 5 | Yes |
x ≥ 5 | x is at least 5 | Yes |
On a number line, an open circle excludes the boundary and a filled circle includes it. Shade left for smaller values and right for larger values. For x ≤ 5, place a filled circle at 5 and shade left.
2. Addition and Subtraction Preserve Order
Worked Example: Solve and Check
Solve x + 4 < 9. Subtract 4 from both sides to get x < 5.
On a number line, use an open circle at 5 and shade left. The value 5 itself is excluded because 5 + 4 = 9 is not less than 9.
Check an included value: x = 2 gives 6 < 9. Check an excluded value: x = 6 gives 10 < 9, which is false.
3. Negative Multiplication Reverses Order
Start with 2 < 5. Multiplying both values by −1 gives −2 and −5. On the number line, −2 is now greater than −5, so the true comparison is −2 > −5. The order reverses because multiplication by a negative reflects positions across zero.
Therefore, when multiplying or dividing both sides of an inequality by a negative number, reverse the inequality symbol. Multiplication or division by a positive number preserves its direction.
Worked Example: Divide by a Negative
Solve −3x + 2 ≥ 14. Subtract 2 to get −3x ≥ 12.
Divide by −3 and reverse the symbol: x ≤ −4.
At x = −4, the original left side is 14, so the boundary is included. At x = −5 it is 17, which also satisfies the inequality.
4. A Context Can Restrict the Solution Set
If a calculation gives n ≤ 6.5 but n counts whole items, only integer values can be used. If a count must also be nonnegative, the permitted values are 0, 1, 2, 3, 4, 5, and 6. State the context rather than rounding 6.5 to 7, which would violate the inequality.
Example: A Capacity Limit
A container holds at most 20 kg. Each identical package weighs 3 kg. For n packages, write 3n ≤ 20.
Then n ≤ 20/3. Since n is a whole-number count, the maximum is 6 packages. Six weigh 18 kg; seven would weigh 21 kg and exceed the limit.
Common Mistakes
Reverse the inequality only when an operation requires it, not simply because a negative number appears.
Do not reverse the symbol when adding or subtracting a negative number.
A strict inequality excludes the boundary even if it is a convenient whole number.
5. Practice
1. Solve 2x + 1 ≤ 9.
2. Solve −4x > 12.
3. Solve 7 − x < 3 and describe its number-line graph.
4. Tickets cost $6 each. With $25 available, what is the greatest number of tickets that can be bought?
Worked Solutions
1. Subtract 1, then divide by 2: x ≤ 4.
2. Divide by −4 and reverse the symbol: x < −3.
3. Subtract 7 to get −x < −4. Divide by −1 to obtain x > 4. Use an open circle at 4 and shade right.
4. Write 6n ≤ 25. The greatest nonnegative integer satisfying this is 4. Four tickets cost $24, while five cost $30.
6. Summary
- An inequality usually describes a set of values.
- Track whether the boundary is included.
- Reverse order when multiplying or dividing by a negative number.
- Apply whole-number or nonnegative restrictions when the context requires them.