An elevator holds at most 1000 kilograms. The operator weighs 80 kilograms and each box weighs 25 kilograms, so with boxes the load is , and you need . There is no single right answer here, only a range of allowed values, and the SAT asks about such ranges in plain algebra and in word problems alike. Solving an inequality means finding that range exactly, edges included.
See also Solving Inequalities on a Number Line in Foundations for Calculus and Sets, Intervals, and Inequalities in Precalculus.
Key terms
Inequality
Definition An inequality compares two real expressions with one of the symbols , , or . For real numbers, means that is positive, and means the same as . A linear inequality in is one that can be written as with , or the same with , or .
In plain words An inequality says which side is bigger. Unlike an equation, it is usually true for a whole stretch of numbers.
Example The inequality is true at , since . It is false at , since is greater than .
Solution set
Definition The solution set of an inequality in is the set of all real numbers that make it true. Two inequalities with the same solution set are equivalent, and solving means replacing the inequality by simpler equivalent ones. These steps keep the solution set.
- Adding or subtracting the same number on both sides.
- Multiplying or dividing both sides by the same positive number.
- Multiplying or dividing both sides by the same negative number and reversing the symbol, so becomes and becomes .
The reversal is needed because, for and , the number is negative, which gives .
In plain words Do the same thing to both sides, as with equations. The one new rule is that a negative multiplier flips the symbol, because but .
Example Solve . Subtracting gives , and dividing by reverses the symbol,
Compare a positive coefficient. In you add and divide by to get , and the symbol stays as it is.
Boundary
Definition The boundary of a linear inequality in is the number where its two sides are equal, the solution of the related equation. For a linear inequality the boundary splits the number line into two pieces, the solutions lie entirely in one of them, and the boundary itself is a solution exactly when the symbol is or .
In plain words Find the place where the two sides tie. Then decide which side of it works and whether the tie point counts.
Example For the related equation gives the boundary . Test one value on each side. At the left side is , and is true, so is included. At it is , and is false, so is excluded. At the boundary the left side is exactly , and is false. The solution set is all .
Strict and inclusive inequalities
Definition The inequalities and are strict, and equality makes them false. The inequalities and are inclusive, where means or . Every legal solving step keeps strict as strict and inclusive as inclusive, even when the symbol reverses.
In plain words The phrases “less than” and “more than” leave the boundary out. The phrases “at most”, “at least”, “no more than” and “no fewer than” let it in.
Example Solve . Subtracting gives , and dividing by gives . The symbol reversed, but it is still inclusive, and works, since .
Context adds one more condition that the algebra does not see. In the elevator problem, gives
A number of boxes must be a whole number, and it cannot be negative, so the allowed values are and the most the elevator can take is boxes, a load of kilograms. Ordinary rounding would turn into , but the inequality decides the direction here. Since must stay at or below , you go down to the nearest whole number, and boxes would give kilograms, over the limit.
Number line
Definition The number line pictures the real numbers in order, with smaller numbers to the left. The solution set of a linear inequality in one variable is drawn as a ray that starts at the boundary and is shaded on the side of the solutions. An open circle at the boundary means it is excluded, and a filled dot means it is included.
In plain words The picture shows the whole answer at once, which side and whether the edge is in.
Example The solution is drawn with an open circle at and shading to the left. The solution gets a filled dot at and shading to the right. In interval notation these are and .
Compound inequality
Definition A compound inequality joins two inequalities with “and” or “or”. The chain is shorthand for and . A chain is solved by applying each step to all three parts at once, reversing both symbols when you multiply or divide by a negative number.
In plain words The word “and” means both conditions must hold. The word “or” means at least one of them must.
Example Solve . Adding to all three parts and then dividing by gives
Check the edges. At the middle is , and holds. At the middle is , and fails.
Intersection
Definition The intersection of two sets is the set of numbers that belong to both. The solution set of two inequalities joined by “and” is the intersection of their separate solution sets. It can be empty, as in and .
In plain words Shade each condition and keep only the part where the shadings overlap.
Example Solve and . The first gives and the second gives . The intersection is , the interval . A value like satisfies the first condition but not the second, so it is out.
Union
Definition The union of two sets is the set of numbers that belong to at least one of them. The solution set of two inequalities joined by “or” is the union of their separate solution sets.
In plain words Shade each condition and keep everything that is shaded at all.
Example Solve or . The first gives and the second gives . The union is . The number satisfies neither part, so it is not a solution, and satisfies the first, which is enough.
Check yourself
- Which of the following is the solution set of ? (A) (B) (C) (D)
Show the answer
The answer is (B). Subtracting gives , and dividing by reverses the symbol, so . Choice (A) forgets to reverse the symbol. Choice (C) adds instead of subtracting it, and choice (D) loses the sign of . As a check, gives , true, while gives , false.
- A club has 250 dollars for a trip. The bus costs 40 dollars, and each T-shirt costs 12 dollars. What is the greatest number of T-shirts the club can buy without going over its budget?
Show the answer
The answer is . The condition is , so and . The number of shirts is a whole number, so the greatest is , which costs dollars. Buying would cost dollars.
- Which value of satisfies or ? (A) (B) (C) (D)
Show the answer
The answer is (D). The first inequality gives and the second gives , and “or” means the union of the two. The value is in the second piece, since . Choice (A) is tempting, but is the boundary of a strict inequality, and is not less than . Choices (B) and (C) lie in the gap between the two pieces.