6th Grade · 6.EE.B.8

Graphing inequalities in 6th grade

Graphing inequalities means showing on a number line every number that makes an inequality true. In 6th grade, your child writes simple inequalities such as $x > 5$ for a real limit and graphs them. Your child also learns that an inequality like this has infinitely many solutions.

What the standard says

The official Common Core wording, quoted exactly. The plain version follows.

6.EE.B.8 Write an inequality of the form x > c or x c to represent a constraint or condition in a real-world or mathematical problem. Recognize that inequalities of the form x > c or x c have infinitely many solutions; represent solutions of such inequalities on number line diagrams.

What it means, in plain words

Inequalities compare two amounts that are not always equal. The sign $>$ means "greater than" and $<$ means "less than". So $x > 5$ says that $x$ is some number bigger than 5, such as 6, 10 or 5.5. Numbers like 5.1 and 5.01 work too, and the list never ends. That is why the inequality has infinitely many solutions.

Writing inequalities starts from a limit in a real situation. "A suitcase must weigh less than 50 pounds" becomes $w < 50$ , where $w$ is the weight in pounds. In 6th grade, the inequality has the letter on one side and a number on the other.

Graphing inequalities in 6th grade uses a number line. Draw an open circle at the number to show that it is not a solution. Then shade the line in the direction of the solutions, with an arrow at the end: to the right for greater than, to the left for less than. Some problems also use $\geq$ (greater than or equal to) or $\leq$ (less than or equal to). Then the circle is filled in, because the number itself is a solution.

Your child should be able to:

Write an inequality for a limit described in words
Explain why an inequality like $x > 5$ has infinitely many solutions
Graph an inequality on a number line with an open circle and shading in the right direction
Read an inequality from its graph, including graphs with negative numbers

Example

A ride is only for people taller than 48 inches. Write and graph an inequality for the heights allowed.

Let $h$ be a rider's height in inches.
"Taller than 48 inches" means greater than 48, so the inequality is $h > 48$ .
Draw an open circle at 48, because a rider who is exactly 48 inches tall cannot ride.
Shade the line to the right of 48. Heights such as 50 and 60 inches are on the shaded part.
The inequality is $h > 48$ , graphed as an open circle at 48 with shading to the right.

Example

Graph $x < -1$ on a number line.

Draw an open circle at $-1$ , because $-1$ is not less than $-1$ .
Numbers less than $-1$ , such as $-2$ and $-5$ , sit to its left.
Shade the line to the left of $-1$ and put an arrow at the end.
Check one number: $-3 < -1$ is true, and $-3$ is on the shaded part.
The graph is an open circle at $-1$ with the line shaded to the left.

Where children usually get stuck

Asked for the solutions of $x > 12$ , your child writes 13.
Equations have one answer, so your child expects one here too. 13 is the first whole number that works.
What helps. Ask for another number that works, and then another. Then try 12.5. The list never ends, which is why the graph is a shaded line, not a single dot.
For $x > -2$ , your child shades toward $-5$ , because 5 is bigger than 2.
Your child compares the digits and ignores the minus signs. On a number line, numbers get smaller as you move left, so $-5$ is less than $-2$ .
What helps. Draw a number line and ask which number is farther to the right. The one farther right is always greater, so $-1$ is greater than $-2$ and $-5$ is less.
Your child writes "fewer than 25 students" as $s > 25$ .
Both signs sit next to the 25, and the direction gets guessed instead of checked.
What helps. Test a number in the words first. Do 30 students fit "fewer than 25"? No. So the inequality must make 30 false, and that is $s < 25$ .

Try four practice questions

Real 6th grade questions from our question bank, easiest first. Answer them right here, check each one, and see a worked solution for any question.

1Warm-up
Look at the inequality below.
$\displaystyle x > 2$
Graph the inequality on the number line.
Tap or click the number line to place your point; a ray appears to the right. Drag the point to aim it left or right, and tap it to switch open or closed.
Show the answer and how to solve it
Answer. 2
The correct graph has an open circle at 2 and a ray pointing to the right.
1. 1The inequality is x > 2. The boundary number is 2.
2. 2Find 2 on the number line. Here, 2 is a labeled tick sitting between 1 and 3.
3. 3The symbol > means numbers greater than 2, so the ray goes to the right. For example, 3 is a solution, but 1 is not. Because the inequality is strict, 2 is not included in the solution set, so use an open circle at 2.
Common mistake. A common mistake is using a closed circle at 2. Because the inequality uses >, not ≥, the boundary value 2 is not part of the solution set and the circle must be open.
The idea underneath. A number line inequality graph has two parts: the number c tells where the boundary point goes, and the inequality symbol tells which side of that point contains the solutions.
2On level
Let t be the temperature in degrees Fahrenheit inside a cold storage room. The room keeps food safe only if its temperature stays below 30 degrees.
Select the matching inequality.
A
$\displaystyle t > 30$
B
$\displaystyle t < 30$
C
$\displaystyle t < 29$
D
$\displaystyle t < 31$
Show the answer and how to solve it
The answer is t < 30 because the temperature must be less than 30 degrees, and the cutoff number stays 30.
1. 1t represents the temperature in degrees Fahrenheit inside the cold storage room. The cutoff value is 30 degrees.
2. 2The phrase stays below 30 degrees means the temperature is less than 30. The boundary number stays 30.
3. 3So the matching inequality is shown below.
Common mistake. A common mistake is changing the boundary to 29 because 29 is one whole number below 30. The inequality must keep 30 as the cutoff and use < to show any temperature less than 30.
The idea underneath. When you translate a word problem into an inequality, keep the boundary number the same and let the comparison words tell you whether the variable must be below it or above it.
3On level
Look at the inequality below.
$\displaystyle x < 3$
Select the graph that matches the inequality.
A
B
C
D
Show the answer and how to solve it
The correct graph for x < 3 has an open circle at 3 with the ray pointing left: open because the inequality is strict, at 3 because that is the boundary number, and pointing left because < means values smaller than 3.
1. 1Start with the inequality shown: x is less than 3.
2. 2The boundary number is 3, so the circle belongs at 3 on the number line.
3. 3The symbol < means values less than 3, so the ray must point left.
4. 4Because the inequality is strict (< not ≤), 3 is not included, so the circle must be open. The correct graph has an open circle at 3 with the ray pointing left — open circle because the inequality is strict, at 3 because that is the boundary number, and pointing left because < means values smaller than 3.
Common mistake. A common mistake is placing the circle at 3 but shading to the right. That graph represents x > 3, which reverses the direction of the inequality.
The idea underneath. A number-line inequality graph is built from three linked ideas: the boundary number, the side chosen by > or <, and whether the boundary is included.
4Stretch
A teacher wrote the statement "x is greater than -6" on the board. Maya built the inequality below.
x
>
6
This equation is read-only.
Select the best diagnosis of the student's work.
A
Reversed > and <
B
Wrong number
C
Used = instead of > or <
D
No mistake
Show the answer and how to solve it
The answer is Wrong number because the statement means x > -6, but Maya wrote x > 6.
1. 1Translate the statement first. The words 'greater than' match a > symbol, and the boundary number is -6, including the negative sign.
2. 2Now compare that target inequality to Maya's work.
3. 3Maya used >, which matches the symbol. But she wrote 6 instead of -6. These are different numbers: -6 is six units to the left of zero, while 6 is six units to the right.
4. 4The comparison symbol is correct, the work is still an inequality, and the only error is the boundary number changing from -6 to 6. The best diagnosis is Wrong number.
Common mistake. A common mistake is picking No mistake because both inequalities use >. But -6 and 6 are different numbers, so x > 6 is not the same as x > -6. The negative sign is part of the boundary number and must be preserved.
The idea underneath. In an inequality, the boundary number matters just as much as the symbol; changing -6 to 6 changes which values satisfy the statement.