Math 1 · Lesson 7 · part 1 of 7
The Words We Use
The vocabulary for this lesson, in one place.
Words we use
| Word | Plain English | What you do |
|---|---|---|
| Absolute value | The distance of a number from zero, written with two bars: |−7| = 7 and |7| = 7. Distance has no direction and is never negative. | Read |x + 2| as: the distance of x + 2 from zero. |
| Less than a distance | |inside| < 5 means the inside is closer to zero than 5: somewhere between −5 and 5. | Set up two problems joined by and: −5 < inside and inside < 5. |
| Greater than a distance | |inside| > 5 means the inside is farther from zero than 5: left of −5 or right of 5. | Set up two problems joined by or: inside < −5 or inside > 5. |
| And | Both facts must be true at the same time. The answer is one stretch of number line between two boundaries. | Write it as a sandwich with x in the middle: −1 < x < 7. |
| Or | At least one fact is true. The answer is two pieces of number line pointing away from each other, with a gap between. | Write the two pieces with or between them: x ≤ −4 or x ≥ 3. Never as a sandwich. |
| Get the bars alone | Anything outside the bars (a number multiplied on, a number added) is peeled off first, using the Lesson 6 moves. | Only split into two problems once the line reads |inside| < number or |inside| > number. |
| Two problems, two answers | Every absolute value inequality becomes two ordinary inequalities with the same inside. | Solve both with the Lesson 6 rules, including the flip if a coefficient (the number on the variable) is negative. |
| No solution | |inside| < a negative number. No distance is that small. | Say no solution. Do not set up two problems. |
| All numbers | |inside| > a negative number. Every distance is bigger than a negative. | Say every number works. |
| Additive inverse | The opposite. Add it to both sides to move a term, inside the bars or outside them. | Step 1 rewrites |x - 3| as |x + (−3)|. |
| Multiplicative inverse | The reciprocal. Multiply both sides by it as the last move. Negative reciprocal: flip the sign. | In an or-problem or an and-problem, a negative coefficient flips both signs. |