Inequality Word Problems: Graphs, Solutions and Mistakes | Grade 6 Math | LevelKite

In this lesson you write and graph inequalities for word problems. You also find out how many solutions an inequality has, and you fix mistakes. You already know the main ideas, so here is a quick review.

An inequality compares two amounts using $<$ , $>$ , $\le$ or $\ge$ . A solution is a value of the variable that makes the inequality true. In $x > 5$ , where $x$ stands for a number, $7$ is a solution because $7 > 5$ is true. The boundary is the number that separates the solutions from the numbers that are not solutions. Here it is $5$ . A graph shows the solutions on a number line, whose ticks are small, evenly spaced marks. A filled circle on the boundary means the boundary is a solution, as in $x \ge 5$ . A hollow circle means it is not, as in $x > 5$ . The ray is the thick part of the line that starts at the circle and runs one way. It covers the side where the other solutions are, and its arrowhead shows that it keeps going.

In a word problem, the rule is often a constraint: a condition that limits a quantity, such as “at most $12$ people.” An inequality says the same thing with a variable and a symbol. To write and graph it:

Name the variable and its unit.
Find the boundary: the number in the constraint.
Decide whether the boundary itself follows the rule. If it does, use a filled circle and a symbol with a bar, $\ge$ or $\le$ . If not, use a hollow circle and $>$ or $<$ .
Do a side test: try one number on each side of the boundary in the story. If the number on the right follows the rule, the solutions lie to the right, so use $>$ or $\ge$ , and the ray points right. If the number on the left follows it, use $<$ or $\le$ , and the ray points left.
Write the inequality. Check that your test numbers make it true or false, just as in the story.
Say which solutions the story allows.

For constraints like these, all the solutions lie on one side of the boundary. That is why one test on each side is enough.

Worked example

An elevator limit

A sign in an elevator says it can carry at most $12$ people. Write and graph an inequality for the number of people the elevator can carry. Then say which solutions make sense in the story.

Name the variable and its unit. Let $p$ be the number of people in the elevator. The unit is people.
Find the boundary. The number in the constraint is $12$ . “At most $12$ ” allows exactly $12$ people, so the boundary is a solution. That calls for a filled circle and a symbol with a bar.
Do the side test in the story. Try $10$ , left of $12$ : ten people fit the limit, so $10$ follows the rule. Try $14$ , right of $12$ : fourteen people are too many. The solutions lie to the left, so the symbol is $\le$ and the ray points left.
Write the inequality and check it with the same test numbers. It is true for $10$ and false for $14$ , just as in the story. So the graph has a filled circle at $12$ and a ray to the left.
$p \le 12 \qquad \text{check: } 10 \le 12 \text{ is true}, \; 14 \le 12 \text{ is false}$
Say which solutions make sense. The ray holds every number that makes $p \le 12$ true, such as $5\frac{1}{2}$ , and it goes on past $0$ into the negative numbers. But people come only in whole numbers, and a count cannot be negative.
$p = 0, 1, 2, \ldots, 12$

Answer

The inequality is $p \le 12$ , graphed with a filled circle at $12$ and a ray to the left. In the story, only the whole numbers from $0$ to $12$ make sense.

The filled circle at

12

and the ray to the left show

p \le 12

, though only the whole numbers from

0

to

12

fit the story.

Worked example

A rain limit

A town closes its ball fields for the day when more than $\frac{1}{4}$ inch of rain falls. Write and graph an inequality for the amounts of rain that close the fields. Then say which solutions make sense.

Let $r$ be the amount of rain that falls in a day, in inches. The boundary is $\frac{1}{4}$ , and the number line below labels its tick $0.25$ . Both name the same number. Multiplying the top and bottom by the same nonzero number, here $25$ , keeps a fraction's value, and $25$ hundredths is written $0.25$ .
$\frac{1}{4} = \frac{1 \times 25}{4 \times 25} = \frac{25}{100} = 0.25$
Exactly $\frac{1}{4}$ inch is not more than $\frac{1}{4}$ inch, so it does not close the fields. The boundary is not a solution, so the circle is hollow.
Side test: $0$ inches, left of the boundary, leaves the fields open. Half an inch, or $0.5$ inch, is right of it. That is more than $\frac{1}{4}$ inch, so it closes the fields. The solutions lie to the right, so the symbol is $>$ and the ray points right. You may write the boundary as $\frac{1}{4}$ or $0.25$ , because they are the same number.
$r > \frac{1}{4} \qquad \text{check: } 0 > 0.25 \text{ is false}, \; 0.5 > 0.25 \text{ is true}$
Say which solutions make sense. Rain amounts do not have to be whole numbers, so fractions and decimals fit the story, such as $0.3$ , $\frac{1}{2}$ and $1\frac{1}{4}$ inches.

Answer

The inequality is $r > \frac{1}{4}$ , which is the same as $r > 0.25$ . Its graph has a hollow circle at $0.25$ and a ray to the right. Any amount of rain greater than $\frac{1}{4}$ inch, whole or not, closes the fields.

The hollow circle at

0.25

, which is

\frac{1}{4}

, and the ray to the right show

r > \frac{1}{4}

.

Now look at an inequality with no story: $x > 2$ , where $x$ can be any number. The graph below has a hollow circle at $2$ , because $2 > 2$ is false. The ray points right, because the test number $3$ makes $3 > 2$ true, while $0$ makes $0 > 2$ false. So how many solutions does $x > 2$ have?

The hollow circle at

2

and the ray to the right show

x > 2

, and the ray keeps going past the last label,

6

.

Key concept

An inequality has infinitely many solutions

When the variable can be any number, an inequality such as $x > c$ , $x < c$ , $x \ge c$ or $x \le c$ , where $c$ is the boundary, has infinitely many solutions. The ray never ends, and every number on it is a solution, including the fractions and decimals between the ticks. A word problem may allow only some of these solutions. In the elevator story, only the whole numbers from $0$ to $12$ made sense.

Mistakes can hide in an inequality or in its graph. An inequality must keep exactly the numbers the story allows, and a graph must show exactly the solutions of its inequality. To check work, look at three parts. If one fails, name the fault and fix it.

Symbol: make sure the boundary is the number in the constraint. Then do side tests in the story and in the inequality. If they disagree, the fault is a reversed symbol.
Circle: ask whether the boundary is a solution. Filled means yes and hollow means no. If the circle gives the wrong answer, the fault is the wrong circle.
Side: test a number on the ray in the inequality. If the result is false, the fault is the ray on the wrong side.

Worked example

Find the fault: a height rule

A water slide sign says riders must be at least $48$ inches tall. Sam lets $h$ be a rider's height in inches. Sam writes $h \le 48$ for the heights that may ride. Find Sam's mistake, name it, and fix it.

Check the boundary first, so the side tests can judge the symbol. The number in the constraint is $48$ , and Sam used $48$ , so the boundary is right.
Do side tests in the story and in Sam's inequality. A rider $50$ inches tall may ride, but $50 \le 48$ is false. A child $40$ inches tall may not ride, but $40 \le 48$ is true. The story and the inequality disagree on both sides, so the fault is a reversed symbol. The word “least” can pull you toward $\le$ , but “at least $48$ ” means $48$ or more.
$50 \le 48 \text{ is false}, \quad 40 \le 48 \text{ is true}$
Fix the symbol by turning it around. Keep the bar, because a rider exactly $48$ inches tall may ride. Recheck: $50 \ge 48$ is true and $40 \ge 48$ is false, which now matches the sign.
$h \ge 48$
Graph the fix. The circle at $48$ is filled because $48$ is a solution. The ray points right because the side test showed the solutions lie to the right. Heights do not have to be whole numbers, so a rider $48\frac{1}{2}$ inches tall may ride too.

Answer

Sam's fault is a reversed symbol. The correct inequality is $h \ge 48$ , graphed with a filled circle at $48$ and a ray to the right.

Sam's fixed graph: the filled circle at

48

and the ray to the right show

h \ge 48

.

Worked example

Find the faults: a fraction boundary

Lee graphs $x \le \frac{1}{3}$ , where $x$ can be any number. Lee draws a hollow circle at $\frac{1}{3}$ and a ray to the right. Find each fault, name it, and describe the correct graph.

Check the circle. It sits at the right number, $\frac{1}{3}$ . The symbol $\le$ means less than or equal to, and $\frac{1}{3}$ is equal to itself, so the boundary is a solution. The circle should be filled, so the hollow circle is the wrong circle.
$\frac{1}{3} \le \frac{1}{3} \text{ is true}$
Check the side. Test $1$ , a number on Lee's ray: $1 \le \frac{1}{3}$ is false, so the ray is on the wrong side. Test $0$ on the other side: $0 \le \frac{1}{3}$ is true, so the solutions lie to the left.
$1 \le \tfrac{1}{3} \text{ is false}, \quad 0 \le \tfrac{1}{3} \text{ is true}$
Describe the correct graph. Split the line from $0$ to $1$ into three equal parts. Each part is $\frac{1}{3}$ of the distance, so $\frac{1}{3}$ is where the first part ends. The graph has a filled circle there and a ray to the left that runs past $0$ and never ends.

Answer

Lee's graph has two faults: the wrong circle and the wrong side. The correct graph of $x \le \frac{1}{3}$ has a filled circle at $\frac{1}{3}$ , one third of the way from $0$ to $1$ , and a ray to the left.

Takeaway: For a word problem, find the boundary, decide whether it is a solution, and use side tests to find the side that holds the other solutions. Then say which solutions the story allows, such as only whole numbers for a count of people. When the variable can be any number, an inequality has infinitely many solutions: the ray never ends, and the numbers between the ticks count too. To check work, test the symbol, the circle and the side. Name any fault you find, then fix it.