On the test
Algebra is about 35% of the Math section, around 6 to 7 questions in each 22-question module. Math modules mix all four domains and go from easier to harder within the module, so Algebra shows up from the first question to the last.
13 to 15 of the 44 questions (about 6 or 7 in each module). All through the module, easier first and harder at the end.
How I approached it
A small story from Aime
When I think about an inequality, I think about a bus with a limit. A minibus can carry at most 18 people. Not exactly 18, but 18 or fewer. If I count 15, it is fine. If I count 19, it is not. The number is not one fixed answer, it is a whole group of numbers that work. An inequality is the same: it describes a group of allowed values. On the SAT I use the same steps as an equation, but I watch the direction of the sign, and I think about what the whole group looks like. Then I choose the one that fits the question.
Inequalities are only equations with one extra rule. Learn that rule, and the questions become very predictable.
What is inside this part
You find all values of x that make a single inequality true.
Flip the sign · Compound inequalities
You turn a limit in a story into an inequality and find the maximum or minimum.
At most and at least · Whole-number answers
You find which points satisfy two inequalities together.
Test a point · Solid and dashed lines
The solution is given and you find a missing number in the inequality.
Match the solution
How to know a question is from this part
A sign like <, ≤, > or ≥, or words that describe a limit.
- The signs <, >, ≤ or ≥ in the question or in the options
- "At most", "at least", "no more than", "minimum", "maximum"
- "Which point is a solution to the system of inequalities"
- "Greatest possible value" or "least possible value"
- "How many integers satisfy"
The lesson
1. What is an inequality?
An inequality compares two things that are not equal. The signs are: < (less than), > (greater than), ≤ (less than or equal to), and ≥ (greater than or equal to).
A solution is any number that makes the inequality true. For x > 4, the numbers 5, 6 and 100 are solutions, but 4 itself is not. For x ≥ 4, 4 is also a solution.
2. Solving one-variable inequalities
Use the same steps as an equation: add, subtract, multiply or divide both sides.
The one new rule: if you multiply or divide both sides by a negative number, you must flip the sign. For example, −3x + 4 ≥ 16 gives −3x ≥ 12. Dividing by −3 flips the sign: x ≤ −4.
Solve −3x + 4 ≥ 16.
Show the steps
- Subtract 4: −3x ≥ 12.
- Divide by −3 and flip the sign: x ≤ −4.
x ≤ −4
3. Word problems
Words tell you the sign. "At most" and "no more than" mean ≤. "At least" and "no fewer than" mean ≥. "Fewer than" means <. "More than" means >.
Often the answer must be a whole number (people, notebooks). Solve, then choose the nearest whole number that still works.
Ange has 21,000 RWF. A bag costs 5,000 RWF and each notebook costs 1,500 RWF. What is the greatest number of notebooks she can buy along with the bag?
Show the steps
- Write 1500n + 5000 ≤ 21000.
- Subtract 5000: 1500n ≤ 16000.
- Divide: n ≤ 10.67.
- She cannot buy part of a notebook, so n = 10.
10 notebooks
4. Compound inequalities
A compound inequality has two parts, like −2 < 3x + 1 ≤ 10. Do the same steps on all three parts at once.
Subtract 1 from all parts: −3 < 3x ≤ 9. Divide by 3: −1 < x ≤ 3. The whole numbers that fit are 0, 1, 2 and 3.
5. Inequalities with two variables
An inequality like y > 2x − 1 has a boundary line y = 2x − 1. The solution is all the points on one side of the line.
If the sign is < or >, the line is dashed and its points are not solutions. If the sign is ≤ or ≥, the line is solid and its points are solutions.
To test a point, put its x and y in all the inequalities. If every one is true, the point is a solution of the system.
Which point satisfies both y > 2x − 1 and y ≤ −x + 5? Choose from (1, 3), (0, −2), (3, 1), (2, 4).
Show the steps
- Test (1, 3): 3 > 1 is true, and 3 ≤ 4 is true. It works.
- Test (0, −2): −2 > −1 is false.
- Test (3, 1): 1 > 5 is false.
- Test (2, 4): 4 > 3 is true, but 4 ≤ 3 is false.
(1, 3)
6. Finding a constant
Sometimes the solution is given, and a letter in the inequality is missing. Solve in terms of the letter, and match to the given solution.
Example: 5 − kx > 11 has solution x < −3. Solve: −kx > 6, and dividing by −k gives x < −6/k when k is positive. Set −6/k = −3, so k = 2.
7. Common traps
Trap 1: forgetting to flip the sign when dividing by a negative.
Trap 2: choosing 11 when the answer is 10.67 and the question asks for the most you can buy. Round down for a maximum.
Trap 3: mixing up < and ≤, and so including or leaving out a boundary value.
Trap 4: testing only one inequality in a system.
See it
| Words | Sign |
|---|---|
| at most, no more than, maximum | ≤ |
| at least, no fewer than, minimum | ≥ |
| fewer than, less than, under | < |
| more than, greater than, over | > |
- 1Do the steps like an equation
- 2Check: did I multiply or divide by a negative?
- 3If yes, flip the sign
- 4For a story, round to a whole number in the allowed direction
- 5Test one number to check
An open circle means the number is not included. A filled circle means it is included.
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