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Expressions and Equations 6-8 / Week 05 / Tuesday
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Week 05 · Inequalities

Tuesday

Write it and graph it
// More than, less than, and a whole set of answers on the number line
⏱ about 20 min

Tuesday: Write It and Graph It

Rocket holds up a puzzle card. "At least 8 cards to enter the Puzzle Post." He frowns. "Is 8 enough?"

"At least 8 means 8 or more," Raven says. "What do you notice that is different from more than 8?"

"Eight counts this time," Rocket says. "So the circle on the 8 should be... filled in?"

Nova hovers above the floor number line, her light settling on the 8. "Filled means included," she says.

Raven colors the paper circle solid and unrolls the ribbon to the right. "x ≥ 8. Eight and everything above."

"And at most 8 would be the other way," Rocket says, "filled circle, ribbon left."

Nova hums. "Four words, four signs. Learn them and every rule becomes a picture."

Four words, four signs

The rule saysSignInequalityCircle at the boundaryArrow
more than 8>x > 8openright
less than 8, fewer than 8<x < 8openleft
at least 8, 8 or more≥x ≥ 8filledright
at most 8, 8 or fewer≤x ≤ 8filledleft

A rule in words becomes an inequality the same way a puzzle became an equation. Name the unknown, then write what the rule says.

"At least 8 cards" says x ≥ 8. Eight itself works, so the circle is filled. "More than 8" says x > 8, and the circle is open.

A number line from 0 to 12 with a filled circle at 8 and an arrow pointing right.

Graph an inequality

  1. Draw a number line that includes the boundary number with room on both sides.
  2. Put a circle at the boundary: open for > or <, filled for ≥ or ≤.
  3. Draw an arrow from the circle toward the numbers that work: right for more, left for less.
  4. Check one number under the arrow by substitution. It should make the inequality true.
WHICH INEQUALITY SAYS THAT?
  • Read the question.
  • Tap your answer.
Rocket needs more than 8 cards. Which inequality says that?
Raven wants at least 8 cards. Which inequality says that?
The jar holds fewer than 30 beans. Which inequality says that?
The walk takes at most 12.5 minutes. Which inequality says that?
READ THE NUMBER LINE
  • Read the question.
  • Tap your answer.
A number line from 0 to 12 with a filled circle at 8 and an arrow pointing rightWhich inequality does this number line show?
A number line from 0 to 8 with an open circle at 3 and an arrow pointing rightWhich inequality does this number line show?
A number line from 0 to 10 with a filled circle at 5 and an arrow pointing leftWhich inequality does this number line show?
A number line from 0 to 6 with an open circle at 2 and an arrow pointing leftWhich inequality does this number line show?
GRAPH X ≤ 5, STEP BY STEP
  • ?Draw a number line from 0 to 10.
  • ?Put a filled circle at 5, because 5 is included.
  • ?Draw an arrow from 5 to the left.
  • ?Check: 3 ≤ 5 is true, so 3 sits under the arrow.
WHY THIS EXERCISEThe circle says whether the boundary counts. The arrow says which side counts. The check proves it.
StatementTrue or false?
"At least 8" is written x ≥ 8.?
"More than 8" uses a filled circle at 8.?
x = 8 makes x ≥ 8 true.?
x = 7 makes x ≤ 5 true.?
8 > 5?
WHY THIS EXERCISEThe sentence x ≥ 8 is true for 8 itself, so the circle is filled. For x > 8 the 8 fails, so the circle is open.
Try it
Write four rules of your own, one for each sign, about beans, cards, steps or minutes.
Under each rule write the inequality and draw its number line.
Draw four number lines, one each for x > 2, x < 2, x ≥ 2 and x ≤ 2. Label the circles open or filled.

Four signs, four pictures. Tomorrow is Puzzle Lab: you tape the number line on the floor and walk the answers.

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