Thursday
Thursday: The Crew's Puzzle Rules
Raven spreads four rule cards on the workbench. "Each one has two steps. Walking the line is slow. Let us undo instead."
Rocket reads card 1. "I have 3 cards done and make x a day for 4 days. I want more than 15. So 4x + 3 > 15."
"What do you notice about the 3?" Raven asks. "It was added. Subtract it from both sides." Rocket writes 4x > 12.
"Then divide both sides by 4," Raven says. "x > 3."
Nova hovers over the card, her light bright. "Same undo as an equation," she says. "Only the sign stays."
"So 4 days of 4 cards works," Rocket says, "and 4 days of 3 does not." He checks: 4 × 3 + 3 = 15. Not more than 15.
Nova hums. "Boundary left out. Open circle."
Solve px + q > r like an equation
A two-step inequality undoes in the same order as a two-step equation. Undo the adding or subtracting first, then the multiplying.
Do the same to both sides at each step. The sign stays as it is, because you only add, subtract, multiply or divide by positive numbers.
The answer is a solved inequality, like x > 3. Graph it with a circle and an arrow, then check one value.
| Inequality | Undo on both sides | Result |
|---|---|---|
| 4x + 3 > 15 | subtract 3 from both sides | 4x > 12 |
| 4x > 12 | divide both sides by 4 | x > 3 |
The crew's rule cards
| Card | Written by | The rule | Inequality | Solved |
|---|---|---|---|---|
| 1 | Raven | Raven has 3 cards done and makes x cards a day for 4 days. She wants more than 15 cards. | 4x + 3 > 15 | x > 3 |
| 2 | Rocket | Rocket has 20 beans and uses 2 beans for each of x puzzles. He wants more than 6 left, so 20 - 2x > 6. | 2x + 6 < 20 | x < 7 |
| 3 | Raven | Raven walks x minutes to the post and 5 minutes back. She has less than 17 minutes. | x + 5 < 17 | x < 12 |
| 4 | Rocket | Rocket stacks x cups a row for 3 rows plus 2 spare cups. He wants at least 14 cups. | 3x + 2 ≥ 14 | x ≥ 4 |
These four rules are the crew's own cards for this week. Card 2 started as 20 - 2x > 6, so Rocket added 2x to both sides first.
That made 20 > 2x + 6, the same as 2x + 6 < 20. The undo steps then gave x < 7.
- Read the question.
- Tap your answer.
- Read the question.
- Tap your answer.
Which inequality does this number line show?
Which inequality does this number line show?
Which inequality does this number line show?| Statement | True or false? |
|---|---|
| x = 4 makes 4x + 3 > 15 true. | ? |
| x = 3 makes 4x + 3 > 15 true. | ? |
| x = 11.5 makes x + 5 < 17 true. | ? |
| To solve 4x + 3 > 15, divide by 4 before subtracting 3. | ? |
Solve it on both sides, graph the answer, and test one number under the arrow.
Four rules undone. Tomorrow you find inequalities in everyday life and review the week.