SHSAT: Inequalities with whole-number answers
Solve linear inequalities, turn the solution into the greatest or least integer the situation allows, and count the integers in a range.
When an inequality gives a decimal answer but the question needs a whole number, how do you know which integer to choose?
Short answer
Solve the inequality, flipping the sign if you divide by a negative. Then round down for a limit, round up to reach a goal, and test the boundary in the words.
Know first: Solving two-step equations, Integers on a number line
Step 1 of 5
Solve like an equation, then pick the integer the story allows
An inequality is solved the same way as an equation, with one extra rule: if you multiply or divide both sides by a negative number, flip the inequality sign.
The solution is usually a range such as x < 10.9. When the answer has to be a whole number of kits, tickets or rounds, the context decides whether you round down or round up.
| Words | Symbol | If you get 10.9, answer | If you get exactly 11, answer |
|---|---|---|---|
| at most, no more than | ≤ | greatest integer: 10 | 11 |
| fewer than, strictly less than | < | greatest integer: 10 | 10 |
| at least, no fewer than | ≥ | least integer: 11 | 11 |
| more than, greater than | > | least integer: 11 | 12 |
Strict signs exclude the boundary itself. Check the boundary number in the words.
Step 2 of 5
Find the budget first, then divide
Worked example: Jerseys on a budget
A team has $200 and must keep $25 for snacks. Jerseys cost $16 each. What is the greatest number of jerseys the team can buy?
- A10
- B11
- C12
- D13
- 1
Find the spending money
200 − 25 = $175 can be spent.
- 2
Write the inequality
16j ≤ 175, so j ≤ 175/16, which is between 10 and 11.
- 3
Choose the integer
Only whole jerseys count, and spending can't pass $175, so round down: 10. Check: 10 jerseys cost $160; 11 cost $176, which is too much.
- 4
See why the other choices are there
11 rounds up past the budget. 12 ignores the $25 reserve. 13 ignores the reserve and also rounds up.
Answer
10 jerseys
Find the wrong step
Find the smallest integer x for which 7 − 3x < −11.
- 1
Subtract 7 from both sides: −3x < −18.
- 2
Divide both sides by −3: x < 6.
What went wrong
Dividing by a negative number flips the sign. The result should be x > 6.
- 3
The integer just below 6 is 5, so the answer is 5.
The fix
x > 6, so the smallest integer is 7. Check: 7 − 21 = −14, and −14 < −11. With x = 5, 7 − 15 = −8, which is not less than −11.
Rounding up and counting
Round up for "at least": with 35 points and 6 points per round, reaching at least 100 needs 6r ≥ 65, so r ≥ 10 5/6. Answer: 11 rounds.
Count integers carefully: −4 ≤ n < 3 includes −4 but not 3, so n = −4, −3, −2, −1, 0, 1, 2. That's 7 integers.
Step 3 of 5
Check the boundary every time
Check yourself · Question 1
How many integers n satisfy −4 < n ≤ 5?
Answer: C
−4 is excluded and 5 is included, so n = −3, −2, −1, 0, 1, 2, 3, 4, 5. That's 9 integers.
Check yourself · Question 2
A phone plan charges a $14 monthly fee plus $3 per gigabyte. What is the greatest whole number of gigabytes that keeps the bill strictly less than $50?
Answer: A
3g + 14 < 50 gives 3g < 36, so g < 12. The greatest integer below 12 is 11. Check: 11 gigabytes cost $47; 12 cost exactly $50, which is not less than $50.
Trap answer: 12
Keeping the boundary
- Why it looks right
- Solving 3g = 36 gives exactly 12, and a clean answer feels finished.
- Why it's wrong
- Strictly less than $50 rules out a bill of exactly $50. The boundary value fails.
- The right answer
- 11, the greatest integer with a bill below $50.
Check yourself · Question 3
An integer n satisfies both 3n − 4 > 8 and 2n + 5 ≤ 23. What is the sum of all possible values of n?
Answer: C
The first gives 3n > 12, so n > 4. The second gives 2n ≤ 18, so n ≤ 9. The integers are 5, 6, 7, 8, 9, and their sum is 35.
Common mistake
I round to the nearest whole number.
- Why it's tempting
- Rounding to the nearest is the rule you use most often in other classes.
- Do this instead
- Let the context decide. A budget or capacity limit means round down; reaching a goal means round up. Then test the integer in the words.
Step 4 of 5
What to remember
Remember
- Solve as you would an equation, but flip the sign when you multiply or divide by a negative.
- Round down for limits such as budgets and capacity, and round up to reach a goal.
- Check the boundary: strict signs exclude it, and both conditions must hold at once.
Step 5 of 5
Practice on a real question
Use what you just learned on this question, then check the explanation.
In your own words
In the practice question, what amount was available to spend, and why did you round in the direction you did?