One for All SAT › SAT notes › Algebra › Linear inequalities in one or two variables
Linear inequalities in one or two variables: SAT Math notes
Writing and solving inequalities like 15s+25p≤120, and deciding which values or points satisfy them.
Study it with worked examples and 8 practice questionsPractise Linear inequalities in one or two variables questions
What this topic tests
An inequality works like an equation, but instead of one exact answer it describes a whole range of answers. 'At most', 'no more than' and 'cannot exceed' all mean ≤. 'At least' and 'no less than' mean ≥. 'More than' and 'less than' are the strict signs, > and <.
You solve an inequality with the same moves you use on an equation. The one new rule: if you multiply or divide both sides by a negative number, flip the sign.
On the SAT you will write inequalities from stories, solve them to find the most or least of something, work with ranges between two limits, and check which points satisfy an inequality or a system of them.
Key ideas
- 'At most' and 'no more than' mean ≤. 'At least' means ≥. 'Exceeds' means >.
- Multiplying or dividing both sides by a negative number flips the inequality sign.
- When the answer must be a whole number, round in the direction the inequality allows: down for a maximum, up for a minimum.
- A range scales. If each tray holds 50 to 60 cookies, 4 trays hold 4(50) to 4(60) cookies.
- To test a point, plug in its coordinates. It is a solution only if the inequality comes out true; for a system, it must make every inequality true.
- With < or >, points on the boundary line are not solutions. With ≤ or ≥, they are.
The 4 kinds of Linear inequalities in one or two variables question
Sorted from every Linear inequalities in one or two variables question in the site's bank: each kind keeps coming up, and has a method that works every time.
1. Write an inequality from a situation
How to spot it. A story with a limit, such as a budget, a goal or a capacity, and answer choices that are inequalities or systems of inequalities. Some versions give you the inequality and ask what a term like 14y means.
How to answer it:
- Name each variable and its units.
- Build an expression for the quantity being limited, such as price times number for each item, added up.
- Turn the limit words into a sign: 'at most' is ≤, 'at least' is ≥, 'more than' is >.
- Put the expression on one side and the limit on the other.
- If there are two limits, such as a number of items and a total cost, write one inequality for each.
The trap. Getting the sign backward, especially with 'at least' and 'no more than'. Say the limit as a sentence: the cost must be at most 80, so cost ≤80. Also check that each price is attached to the right variable.
2. Find the greatest or least possible value
How to spot it. The question asks for the maximum or minimum number of hours, items, trips or people, or for all the values of x that make one option cost more than another.
How to answer it:
- Write the inequality: the total cost or amount on one side, the limit on the other.
- Solve it like an equation. Flip the sign if you multiply or divide by a negative.
- If the answer must be a whole number, round the right way: a maximum rounds down, a minimum rounds up.
- Check by plugging in your answer and the next whole number past it. One should work and the other should not.
The trap. Rounding the wrong way. If n≤16.1, the greatest whole number is 16, not 17. If n>27.1, the least whole number is 28. Also remember any one-time fee, or money already earned, before you apply the limit.
3. Ranges between two limits
How to spot it. The story gives a low and a high value, such as 'between 72 and 77 miles a day' or 'at least 7.5 and no more than 9 feet', and asks for a value that fits or for the range of a total.
How to answer it:
- Write the range as a compound inequality: low ≤ quantity ≤ high. Use < instead if the ends are not included.
- Do the same thing to all three parts. Multiply both ends by a count, or put both ends through the formula.
- Add any fixed starting amount to both ends.
- If you divide a fixed total by the quantity, the largest quantity gives the smallest result, so the ends swap places.
- Match your range to the choices, checking whether the ends are included.
The trap. Adding when you should multiply. If a whale travels 72 to 77 miles each day for 16 days, the total runs from 72(16) to 77(16), not from 72+16 to 77+16. And when you divide a fixed total by a range, remember that the ends flip.
4. Check points and regions
How to spot it. You are given an inequality, or a system of inequalities, in x and y, and asked which point, table or region is part of the solution set.
How to answer it:
- Plug each choice's x and y into the inequality.
- Keep a choice only if the statement comes out true. For a system, it must be true for every inequality.
- If a choice fails the first inequality, cross it out; you don't need to test it any further.
- For questions about a region or the possible values of y, use the limit on x to work out the limit on y, or test a sample point from each region.
- Desmos shortcut: type the inequalities to shade the solution region, then plot the points and see which one lands in the overlap.
The trap. Counting a point on the boundary line when the sign is strict. If the inequality is < or >, a point that makes both sides equal is not a solution.
Practise Linear inequalities in one or two variables
On One for All SAT every question has a step-by-step solution, worked both by hand and in Desmos, the calculator built into the digital SAT, and the notes have worked examples of each kind of question, then practice questions you can check as you go. It's free, and there's nothing to sign up for.
Open the Linear inequalities in one or two variables notes