Chapter 6 · Equations and inequalities

Writing Inequalities for Real-World Situations

Turn real-world limits into inequalities by choosing the variable, the direction and whether the boundary counts. Then state which values make sense in the situation.

12 min read

Many real situations set a limit instead of naming one exact number. You can spend at most 40 dollars at a fair. A team needs at least 9 players to start a game. Many values fit each rule. In this lesson, you will write one short math statement that describes all of them.

An inequality is a math statement that compares two quantities, or amounts, using one of four symbols: $<$ , $>$ , $\le$ or $\ge$ . The limit number in a situation, such as 40 or 9, is called the boundary. It marks where the allowed values start or stop.

First, make sure you know exactly what each symbol says. In the table, the letter $x$ is a variable, a letter that stands for a number. Here $x$ can be any number, and the boundary is 10.

The four inequality symbols with a boundary of 10

Inequality	In words	Is the boundary included?	Some values that work
$x < 10$	$x$ is less than 10	No, because $10 < 10$ is false	$9.5$ , $4$ , $0$ , $-3$
$x > 10$	$x$ is greater than 10	No, because $10 > 10$ is false	$10.5$ , $11$ , $40$
$x \le 10$	$x$ is less than or equal to 10	Yes, because $10 \le 10$ is true	$10$ , $9.5$ , $0$ , $-3$
$x \ge 10$	$x$ is greater than or equal to 10	Yes, because $10 \ge 10$ is true	$10$ , $10.5$ , $40$

The table shows two things about every symbol.

Direction. $<$ and $\le$ allow values smaller than the boundary and no values larger than it. $>$ and $\ge$ allow values larger than the boundary and no values smaller than it.
The boundary itself. The extra bar in $\le$ and $\ge$ means or equal to, so these two symbols include the boundary. The symbols $<$ and $>$ exclude it.

You can test the boundary by substituting, which means putting a number in place of the variable. Put 10 in place of $x$ . The statement $10 \le 10$ is true because 10 equals 10. The statement $10 < 10$ is false because 10 is not smaller than itself.

Situations use everyday words instead of symbols. The next table matches common phrases to the four symbols. It works when a sentence compares a quantity with a limit in this order: the quantity is [phrase] the boundary, as in the cost is at most 40. Write the quantity on the left and the boundary on the right.

If a phrase is not in the table, ask two questions yourself: must the quantity be larger or smaller than the boundary, and may it equal the boundary?

Phrases that compare a quantity with a boundary

Phrase	What it tells you	Symbol	Is the boundary included?
more than, greater than, above	larger than the boundary	$>$	No
less than, fewer than, below	smaller than the boundary	$<$	No
at least, no less than, no fewer than	the boundary or larger	$\ge$	Yes
at most, no more than, no greater than	the boundary or smaller	$\le$	Yes

Key concept

Four decisions turn a situation into an inequality

Make these decisions in order.

Variable. Choose a letter for the number that can change, and name its unit.
Quantity and boundary. Use your variable to write the quantity the situation limits. The limit number is the boundary.
Direction. Must the quantity be larger or smaller than the boundary?
Boundary. If the quantity may equal the boundary, use $\le$ or $\ge$ . If it may not, use $<$ or $>$ .

Then state which values of the variable make sense in the situation. Use only whole numbers when you count things. Use only nonnegative numbers (zero or positive) when a value cannot be below zero. Some situations, such as temperatures, need neither limit.

Check what you write. Test only values of the variable that make sense in the situation. A value that follows the situation's limit must make your inequality true. A value that breaks the limit must make it false. A value that makes the quantity equal to the boundary must make the inequality true if you used $\le$ or $\ge$ and false if you used $<$ or $>$ .

Worked example

One quantity with a negative boundary

A ski resort can make snow only when the air temperature is below $-2$ degrees Celsius. Write an inequality for the temperatures at which the resort can make snow.

Choose the variable. The air temperature can change, so let c be the air temperature in degrees Celsius.
Find the quantity and the boundary. The rule limits the temperature itself, so the quantity is c. The limit is negative 2 degrees, so the boundary is negative 2.
Decide the direction. Below means colder, and a colder temperature is a smaller number, so the temperature must be smaller than the boundary.
Decide about the boundary. A temperature of exactly negative 2 degrees is not below negative 2 degrees, so the boundary is excluded. Use the less-than symbol.
$c < -2$
State which values make sense. Temperatures can be negative and do not have to be whole numbers, so c is not limited to whole or nonnegative values here. For example, negative 3.5 degrees is a real temperature that follows the rule.
Check. At negative 5 degrees the resort can make snow, and negative 5 is less than negative 2, so the inequality is true. At 0 degrees it cannot, and 0 is not less than negative 2, so the inequality is false. Negative 2 is not less than itself, so the boundary is left out, as the rule says.
$-5 < -2$

Answer

The resort can make snow when $c < -2$ , where $c$ is the air temperature in degrees Celsius. In this situation, $c$ can be negative and does not have to be a whole number.

Worked example

A total cost that depends on a count

You can spend at most 40 dollars at a fair. Admission costs 7 dollars, and each ride ticket costs 3 dollars. Write an inequality for the number of ride tickets you can buy.

Choose the variable. The number of ride tickets can change, so let r be the number of ride tickets you buy.
Write the quantity the situation limits, which is the total cost in dollars. Each ticket costs 3 dollars, so r tickets cost 3 times r dollars. Add the 7 dollars for admission, which you pay once.
$3r + 7$
Choose the symbol. The boundary is 40. At most 40 means the total cost must be 40 or smaller, and spending exactly 40 dollars is allowed. The boundary is included, so use the less-than-or-equal-to symbol.
$3r + 7 \le 40$
State which values make sense. You count tickets, and you cannot buy part of a ticket or a negative number of tickets, so r must be a whole number. A number such as 2.5 makes the inequality true, because the total would be 14.5 dollars and 14.5 is less than 40, but you cannot buy half a ticket.
$3 \cdot 2.5 + 7 = 7.5 + 7 = 14.5$
Check with whole numbers. With 11 tickets the total cost is exactly 40 dollars, and 40 is less than or equal to 40, so the inequality is true. With 12 tickets the total cost is 43 dollars, which is more than 40, so the inequality is false. Both results match the situation.
$3 \cdot 11 + 7 = 33 + 7 = 40 \qquad 3 \cdot 12 + 7 = 36 + 7 = 43$

Answer

The situation is represented by $3r + 7 \le 40$ , where $r$ is the number of ride tickets you buy and $r$ is a whole number.

Worked example

A quantity that shrinks

A candle is 20 centimeters tall when it is lit. It burns down at a steady rate of 2 centimeters each hour. Write an inequality for the times when the candle is taller than 6 centimeters.

Choose the variable and write the quantity. Let t be the time since the candle was lit, in hours. The situation limits the height in centimeters. Burning down is a decrease, so subtract: in t hours the candle loses 2 times t centimeters from its starting height of 20 centimeters.
$20 - 2t$
Choose the symbol. The boundary is 6. Taller than 6 means the height must be larger than 6. A candle that is exactly 6 centimeters tall is not taller than 6 centimeters, so the boundary is excluded. Use the greater-than symbol.
$20 - 2t > 6$
State which values make sense. Time since lighting cannot be negative, so t must be nonnegative. The candle burns steadily between whole hours, so t does not have to be a whole number.
Check with times that make sense. After 6.5 hours the height is 7 centimeters, and 7 is greater than 6, so the inequality is true. After 7 hours the height is exactly 6 centimeters, and 6 is not greater than 6, so the inequality is false. The boundary is left out, as it should be.
$20 - 2 \cdot 6.5 = 20 - 13 = 7 \qquad 20 - 2 \cdot 7 = 20 - 14 = 6$

Answer

The candle is taller than 6 centimeters when $20 - 2t > 6$ , where $t$ is the number of hours since it was lit and $t$ is nonnegative.

Takeaway: A situation with a limit becomes an inequality when you name the variable and its unit, write the quantity the situation limits and compare it with the boundary. Choose the direction, then decide about the boundary: $<$ and $>$ leave it out, while $\le$ and $\ge$ keep it in. Finish by stating which values make sense in the situation, such as whole numbers for a count or nonnegative numbers for a time that cannot be below zero.

Next, you will solve one-step inequalities, which means finding every value that makes them true, and graph those values. There you will meet one more condition: multiplying or dividing both sides of an inequality by a negative number reverses the symbol.

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Solve and Graph One-Step Inequalities

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