Linear inequalities and the flip
An inequality is an equation with a direction attached. Every step you already know still works, and exactly one step changes the answer: multiplying or dividing both sides by a negative number reverses the sign. Everything else in the topic is reading the question precisely.
Solve it like an equation, then check the direction
Add, subtract, and multiply by positive numbers exactly as you would with an equals sign. The solution comes out as a range rather than a point. For 3x - 5 < 10, adding 5 and dividing by 3 gives x < 5. Nothing about the direction changed, because both operations used positive numbers.
The flip, and why it happens
Multiplying or dividing both sides by a negative number reverses the inequality. From -2x > -12, dividing by -2 gives x < 6, not x > 6. The reason is that multiplying by a negative mirrors the number line: a larger value becomes smaller, so the order of the two sides swaps. Dividing by a positive number never flips, and neither does adding or subtracting anything.
- Add or subtract any number: direction unchanged
- Multiply or divide by a positive number: direction unchanged
- Multiply or divide by a negative number: direction reverses
- Swap the two sides completely: direction reverses
Borderline words in the question
The phrase greatest integer means the largest whole number inside the solution, and least integer means the smallest. After solving x < 6, the greatest integer is 5, because 6 itself is excluded. The phrase at most means less than or equal, and at least means greater than or equal, so a plan that costs at most 40 dollars is written with a less than or equal sign, not a strict one.
Word problems with a budget or a limit
Two shapes cover most of them. A fixed cost plus a rate per unit is an expression such as 15 + 4n, which you compare with the limit. A capacity problem is the same with a ceiling instead of a floor. Define the variable with its unit, write the comparison, then solve and remember that a number of people or boxes must be a whole number, which is often where the last half step lives.
Two inequalities at once
A system of inequalities is solved by graphing and looking at the overlap, or by testing the answer choices as points. Testing points is usually faster on the test: take each option, substitute into both inequalities, and keep the one that satisfies both. The shaded region is the set of all points that work, so a question about a point inside the region is a substitution question.
When do you flip the inequality sign?
When you multiply or divide both sides by a negative number. Adding, subtracting, or multiplying by a positive number never flips it.
What does greatest integer value mean?
The largest whole number that still satisfies the inequality. If the solution is x < 6, the greatest integer is 5, because 6 is not included.
How do I know whether to use less than or less than or equal?
At most means less than or equal, at least means greater than or equal. Strictly less than is used when the boundary itself does not count, as with fewer than 10 people.
How do I solve a system of inequalities?
Either graph both and take the overlap, or substitute each answer choice into both inequalities. Substitution is usually quicker when the options are points.
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