Back to Write an inequality to represent a constraint or condition in a real-world or mathematical problem

Write an Inequality to Represent a Constraint or Condition

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Grade 6·21 problems·~35 min·Common Core Math - Grade 6·standard·6-ee-b-8

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Recall / Warm-Up

Which inequality symbol means 'is greater than'?

≤

Which number is greater: −2 or 1?

−2, because it is farther from zero.

1, because it is to the right of −2 on a number line.

−2, because negative numbers are larger than positive.

They are equal.

On a number line, which direction do larger numbers appear?

To the left.

To the right.

In the middle.

Above zero.

Fluency Practice

A box is shorter than 6 inches. Let $h$ represent the height. Which inequality represents this constraint?

$h > 6$

$h = 6$

$h < 6$

$6 < h$

You must score more than 80 points to pass. Let $s$ represent your score. Which inequality represents this requirement?

$s < 80$

$s = 80$

$s > 80$

$80 > s$

The temperature is colder than −10°C. Let $t$ represent the temperature. Which inequality is correct?

$t > -10$

$t < -10$

$t = -10$

$t > 10$

Which of the following values satisfies the inequality $x > 5$ ?

$x = 3$

$x = 5$

$x = 5.1$

$x = 0$

For the inequality $n < 3$ , how many values satisfy it?

Exactly 2 values.

Exactly 3 values (0, 1, 2).

Infinitely many values — all numbers less than 3.

Only the integer values less than 3.

Varied Practice

An elevator can hold fewer than 10 people. Let $p$ represent the number of people. Which inequality represents this?

$p > 10$

$p = 10$

$p \leq 10$

$p < 10$

Write the inequality for each constraint.

'You have more than 20 dollars' → ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲

'The box weighs less than 5 pounds' (let $w$ = weight) → ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲

more than 20 dollars:

less than 5 pounds:

The inequality $x > 3$ has infinitely many solutions. Which statement about the solution set is correct?

The solutions are only the integers 4, 5, 6, 7, ...

$x = 3$ is the only solution.

The solutions include all numbers to the right of 3 on the number line — not just integers.

There are exactly 3 solutions.

Which number line graph correctly shows the solutions to $x > 2$ ?

Open circle at 2, arrow pointing left.

Closed (filled) circle at 2, arrow pointing right.

Open circle at 2, arrow pointing right.

Closed (filled) circle at 2, arrow pointing left.

$Number line from −6 to 4 with an open circle at −3 and an arrow pointing left.$

A number line shows an open circle at −3 with an arrow pointing left. Which inequality does this represent?

$x > -3$

$x = -3$

$x < -3$

$x \geq -3$

Word Problems / Application

A bridge has a weight limit. Only vehicles lighter than 4 tons may cross. Let $w$ represent a vehicle's weight in tons. Which inequality represents the constraint, and which vehicle may cross?

$w < 4$ ; a vehicle weighing 3.9 tons may cross.

$w > 4$ ; a vehicle weighing 5 tons may cross.

$w < 4$ ; a vehicle weighing exactly 4 tons may cross.

$w \leq 4$ ; a vehicle weighing 4 tons may cross.

A park entrance requires visitors to be more than 48 inches tall to enter the water slide. Let $h$ represent a visitor's height in inches.

Write the inequality for this constraint.

$h < 48$

$h > 48$

$h = 48$

$h < 0$

Which number line graph correctly shows the solutions?

Open circle at 48, arrow pointing right.

Open circle at 48, arrow pointing left.

Closed circle at 48, arrow pointing right.

Closed circle at 48, arrow pointing left.

Error Analysis

Priya solved an inequality and tested $x = 10$ for $x < 10$ . She found $10 < 10$ is false. She then concluded: "Therefore, the solution is all numbers greater than 10."

What error did Priya make?

She found the wrong boundary value.

She tested the boundary value correctly but drew the wrong conclusion. $10 < 10$ being false means 10 is NOT a solution, but values LESS than 10 (like 9, 8, 7, ...) satisfy $x < 10$ .

She should have used a closed circle.

Her conclusion is correct.

$Prompt card showing Dante's graph for x greater than 3 with a filled circle at 3 and a right-pointing arrow, plus prompts asking the student what should be checked.$

Dante graphed the solutions to $x > 3$ on a number line. He drew a closed (filled) circle at 3 with an arrow pointing right.

What error(s) did Dante make?

The arrow direction is wrong.

The circle type is wrong — it should be open.

Both the circle type and arrow direction are wrong.

He made no errors.

Marcus graphed the solutions to $x < 5$ on a number line. He drew an open circle at 5 with an arrow pointing right.

What error did Marcus make?

He used a closed circle instead of an open circle.

The arrow should point left, not right.

He placed the circle at the wrong number.

He made no errors.

Challenge

Explain in your own words why $x > 4$ has infinitely many solutions, but $x + 4 = 9$ has only one solution. Use at least one example for each.

A number line shows an open circle at 7 with an arrow to the left AND an open circle at 2 with an arrow to the right. What situation does this represent?

$x < 7$ and $x > 2$ , meaning $x$ is between 2 and 7 (not including 2 or 7).

$x > 7$ and $x < 2$ , meaning $x$ is outside the range.

$x = 7$ or $x = 2$ .

$x < 2$ or $x > 7$ .

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