Math 1 · Lesson 6 · part 1 of 6
The Words We Use
The vocabulary for this lesson, in one place.
Words we use
| Word | Plain English | What you do |
|---|---|---|
| Inequality | A sentence that says one side is bigger than the other, or at least as big. It uses one of the four signs: <, >, ≤, ≥. | Solve it the same way as an equation, with one extra rule at the end. |
| Gator eats greater | The open side of the sign is a mouth, and the mouth always faces the bigger number: 7 > 3, and 3 < 7. | Read every sign this way, and you will never confuse < with >. |
| Less than, greater than | x < 5 says x is smaller than 5. x > 5 says x is bigger than 5. Neither one includes the 5 itself. | On a number line, draw an open circle at the boundary: the 5 is not part of the answer. |
| Or equal to | The little bar under the sign. x ≤ 5 means x is smaller than 5 or exactly 5. x ≥ 5 means bigger than 5 or exactly 5. | Draw a closed circle at the boundary: the 5 is included. |
| Solution | Every number that makes the sentence true. An equation usually has one solution. An inequality usually has a whole side of the number line. | Write the answer as x < 5 (not as one number), and shade the number line. |
| Boundary | The number where the answer starts or stops, like the 5 in x < 5. | Test it in the check. It is included only when the sign has the or-equal-to bar. |
| Addition property of inequality | Add the same thing to both sides, and the sign stays the same. This includes adding a negative number. | Use it to move plain numbers and x terms, exactly as in Lesson 5. Adding never flips the sign. |
| Multiplication property of inequality (positive) | Multiply both sides by the same positive number, and the sign stays the same. | Multiplying by a positive multiplicative inverse (the reciprocal) is safe: 13, 2, 32. |
| Multiplication property of inequality (negative) | Multiply both sides by the same negative number, and the sign turns around. 2 < 6 becomes −6 > −18 when you multiply by −3. | This is the only new rule. It fires when the coefficient (the number on the variable) is negative, because its reciprocal is negative too. |
| Turning the sentence around | 3 > x and x < 3 are the same sentence read from the other end. The gator still eats the 3. | Put x first when you write the answer. This is not a flip: the sign changes direction only because the sides traded places. |
| Additive inverse | The opposite. The opposite of 7 is −7, and the opposite of 2x is −2x. | Add it to both sides to make a term go away. It makes the additive identity (the do-nothing number 0) appear. |
| Multiplicative inverse | The reciprocal, the flipped-over number. The reciprocal of −2 is −12. A negative number has a negative reciprocal. | Multiply both sides by it as the last move. If it is negative, flip the sign. |
| Coefficient | The number on the variable. In (−2)x it is −2. A lonely x is 1x, and −x is (−1)x. | Look at its sign before the last move: a negative coefficient means a flip is coming. |