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Algebra 1 9-12 / Week 02 / Thursday
4/6
Week 02 · Equations and Inequalities in One Variable

Thursday

Inequalities for a limit
// Keep both sides equal, and know why
⏱ about 20 min

Thursday: Inequalities for a Limit

Comet reads the Maker Fair rules she wrote with Wren. "Every build on the table stays under two kilograms."

"Two thousand grams," Wren says. "The base plate is 350 grams on our kitchen scale. Each bracket is 120."

"So 120n + 350 has to be at most 2000," Comet says. "That is not an equation. It is a limit."

Nova projects a number line along the bench. "Would you like a hint? Solve it like an equation, then read the sign."

"Subtract 350 from both sides. 120n ≤ 1650. Divide by 120. n ≤ 13.75."

"Brackets come whole," Wren says. "What do you notice? Thirteen brackets fit. Fourteen is too heavy."

"An inequality gives a whole range of answers," Comet says. "And the build picks the one that makes sense."

An inequality is a limit

An inequality uses <, >, ≤ or ≥ instead of an equals sign. Its solutions are usually a whole range of numbers.

The mass rule: 120n + 350 ≤ 2000, in grams. Solve it with the same moves as an equation.

Subtract 350 from both sides, then divide by 120. The result is n ≤ 13.75.

Brackets come in whole numbers, so the most the crew can add is 13. Check: 120 × 13 + 350 = 1,910 grams.

A number line from 9 to 18 with a filled circle at 13.75 and an arrow pointing left: n ≤ 13.75.

One sign to watch

Adding, subtracting, or multiplying and dividing by a positive number keeps the inequality sign as it is.

Multiplying or dividing both sides by a negative number flips the sign. 3 < 5, but -3 > -5.

The time rule: 30 - 2x > 12, minutes left in a slot after x two-minute runs. Subtract 30: -2x > -18. Divide by -2 and flip: x < 9.

The crew's Loft limits

Here are three limits the crew wrote for the fair. Every number is their own, made up for the story.

Limit in wordsInequalitySolutionWhole numbers that fit
Plate plus brackets at most 2000 grams120n + 350 ≤ 2000n ≤ 13.750 to 13
Fins plus nose at least 40 grams6f + 10 ≥ 40f ≥ 55 or more
More than 12 minutes left after the runs30 - 2x > 12??
SOLVE THE LIMITS
  • Read the question.
  • Tap your answer.
The mass rule is 120n + 350 ≤ 2000. Which values of n keep the build under the limit?
The fin rule is 6f + 10 ≥ 40. Which values of f make it true?
The time rule is 30 - 2x > 12. Which values of x make it true? Watch the sign.
Which number of brackets keeps 120n + 350 ≤ 2000 true: 12, 14 or 15?
The mass rule 120n + 350 ≤ 2000: what is the most brackets, a whole number, the crew can add?
The fin rule 6f + 10 ≥ 40: what is the fewest fins, a whole number, that works?
The time rule 30 - 2x > 12: what is the most runs, a whole number, that leaves more than 12 minutes?
WRITE THE INEQUALITY
  • Read the question.
  • Tap your answer.
A shelf holds at most 2,500 grams. The box is 400 grams and each part is 150. Which inequality fits?
A launcher needs at least 4 fins. Which inequality says so?
Fewer than 10 runs fit in the slot. Which inequality says so?
StatementTrue or false?
n = 13 satisfies 120n + 350 ≤ 2000.?
n = 14 satisfies 120n + 350 ≤ 2000.?
f = 5 satisfies 6f + 10 ≥ 40.?
Dividing both sides of an inequality by -2 flips the sign.?
An inequality has exactly one solution, like an equation.?
WHY THIS EXERCISEThe crew's limits are models. Reading them right keeps the fair table safe and the launcher steady.
Draw the number line for n ≤ 13.75 and mark the whole numbers of brackets that fit.

Sharp limit-reading. Tomorrow you rearrange whole formulas for the letter you need, then review the week.

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