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Infinitely many solutions, and what an open or filled circle means.
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In this lesson you write and graph inequalities such as $x > 3$. The change from equations is real: an equation usually has one answer, an inequality has infinitely many, and you show them as a region rather than a point. The open or filled circle carries genuine information: whether the boundary value is itself a solution. You also turn everyday rules like 'at least 48 inches tall' into inequalities.
You can place whole numbers, fractions, decimals and negative numbers on a number line, and you know that numbers get bigger as you move right. You can compare two numbers with the symbols $<$ and $>$, as in $-3 < 2$. You can also test whether a number makes an equation true, by substituting it. This lesson uses the same symbols to describe not one number but a whole range of numbers.
| Term | What it means |
|---|---|
| Inequality | A statement that one quantity is less than, greater than, or not equal to another, such as $x > 3$. |
| Solution of an inequality | A number that makes the inequality true when you substitute it. |
| Boundary | The number where the solutions start or stop, such as 3 in $x > 3$. |
| Open circle | A hollow circle on a number line: the boundary is not a solution. Used for $<$ and $>$. |
| Filled circle | A solid circle on a number line: the boundary is a solution. Used for $\le$ and $\ge$. |
| At least / at most | At least 10 means $\ge 10$; at most 10 means $\le 10$. Both include 10. |
| Whole numbers | The numbers 0, 1, 2, 3 and so on, with no fractions and no negatives. |
An equation like $x = 3$ has exactly one solution. An inequality like $x > 3$ has infinitely many: 4, 5, 10, 1,000, and also 3.5, 3.01 and $3\frac{1}{8}$. Every number greater than 3 works. You could never list them all, so we draw them.
To graph $x > 3$ on a number line, put a circle at the boundary, 3, and shade the part of the line where the solutions are. Greater means to the right, so the shading runs right and an arrow shows that it never stops.
The circle carries real information. 3 is not greater than 3, so 3 is not a solution of $x > 3$, and we draw an open (hollow) circle. For $x \ge 3$, read 'x is greater than or equal to 3', the number 3 is a solution, and we draw a filled circle. The little line under $\ge$ and $\le$ is half of an equals sign, and it is exactly what makes the boundary count.
There are four symbols to know:
| Symbol | Read as | Circle | Shading |
|---|---|---|---|
| $x < 3$ | less than | open | left |
| $x > 3$ | greater than | open | right |
| $x \le 3$ | less than or equal to | filled | left |
| $x \ge 3$ | greater than or equal to | filled | right |
Another way: picture
Imagine the number line as a road and the boundary as a gate. For $x > 3$ the gate is closed at 3 and everything past it to the right is open road. An open circle is a hole in the road: you can get as close to 3 as you like, 3.1, 3.01, 3.001, but you can never stand on 3 itself. A filled circle is a solid stepping stone you are allowed to stand on.
Another way: story
A sign at a water slide says 'Riders must be at least 48 inches tall'. A rider who is exactly 48 inches may ride, and so may anyone taller: 50 inches, 60, 72. That is $h \ge 48$, with a filled circle at 48 and shading to the right. A child who is 47 inches tall, just left of the circle, has to wait a year.
Everyday rules are often inequalities in disguise. The key is to decide two things: does the boundary number count, and do the allowed values go up or down?
| Words | Boundary counts? | Direction | Symbol |
|---|---|---|---|
| more than, greater than, above, over | no | up | $>$ |
| less than, fewer than, below, under | no | down | $<$ |
| at least, no less than, minimum | yes | up | $\ge$ |
| at most, no more than, maximum, up to | yes | down | $\le$ |
Start by choosing a letter for the quantity, then write the letter first. 'You may bring at most 3 guests' becomes $g \le 3$. 'The temperature stayed above $-5$ degrees' becomes $t > -5$.
Be careful with 'at least' and 'at most'. They sound like 'less' and 'more', but they point the other way. 'At least 10' means 10 or more, so it is $\ge 10$. 'At most 10' means 10 or less, so it is $\le 10$. If you are unsure, test a number: does 11 satisfy 'at least 10'? Yes, so the solutions go up.
To check whether a number is a solution, substitute it and ask whether the statement is true. Is 7 a solution of $x \le 9$? $7 \le 9$ is true, so yes. Is 9 a solution? $9 \le 9$ is true, because 9 equals 9 and the symbol allows 'equal to'. Is 9.5? $9.5 \le 9$ is false, so no.
Testing also works when the inequality has an expression in it, such as $x + 4 > 10$. Try $x = 5$: $5 + 4 = 9$, and $9 > 10$ is false. Try $x = 8$: $12 > 10$ is true. The boundary is where the two sides are equal: $x + 4 = 10$ when $x = 6$. So the solutions are $x > 6$.
Testing is also the best way to check a graph. Pick a number in the shaded part and make sure it works. Pick one outside the shading and make sure it does not. If you get the opposite, the shading is on the wrong side.
Sometimes the variable counts things that cannot be split: people, tickets, boxes. Then only whole numbers are real answers, even though the inequality itself allows fractions.
Suppose a van has room for fewer than 8 passengers: $p < 8$. The numbers 7.5 and $6\frac{1}{2}$ are solutions of the inequality, but half a person cannot ride. The sensible answers are 0, 1, 2, 3, 4, 5, 6 and 7, and there are 8 of them, because 0 counts too. On a number line you would draw dots at each of those whole numbers instead of shading.
So always read the question as well as the symbol. It tells you whether the answer is a shaded ray of every number, or a list of separate whole numbers.
An inequality does not always start with the letter. The statement $10 > x$ is perfectly good: it says 10 is greater than $x$. But it is easy to misread. The safest move is to rewrite it with the letter first. If 10 is greater than $x$, then $x$ is less than 10, so $10 > x$ means the same as $x < 10$.
Notice that the symbol still points at the same number. The narrow end of $<$ or $>$ always points to the smaller side, and the open mouth faces the larger side, whichever way round you write it. In $10 > x$ the mouth faces 10; in $x < 10$ it still faces 10.
The same goes for 'or equal to'. $-2 \le y$ is the same as $y \ge -2$: a filled circle at $-2$ and shading to the right. Swapping the two sides and turning the symbol around together never changes which numbers are solutions.
Step 1: Name the variable and decide what it stands for.
Step 2: Find the boundary, the number where the rule switches from true to false.
Step 3: Choose the symbol. Up or down? Does the boundary count? Use the table of words if the rule is written in a sentence.
Step 4: Draw the circle. Filled for $\le$ or $\ge$; open for $<$ or $>$.
Step 5: Shade. Right for greater, left for less, with an arrow to show the solutions go on forever. If only whole numbers make sense, draw dots instead.
Step 6: Check with two test values: one in the shaded part (it must make the inequality true) and one outside it (it must make the inequality false). Also test the boundary itself to confirm the circle.
A useful habit: read the finished inequality aloud with the letter first. '$x$ is greater than or equal to 3' tells you right away that the answers are 3 and bigger, so the graph starts with a filled circle and runs right.
The United States Constitution sets minimum ages for federal offices. A member of the House of Representatives must be at least 25 years old, a senator at least 30, and the president at least 35. For the president's age $a$ in years, the rule is
$$a \ge 35.$$
On a number line that is a filled circle at 35, because a 35-year-old is allowed, and shading to the right, because there is no maximum age. A 34-year-old, just to the left of the circle, is not eligible.
Compare the three offices on one number line: filled circles at 25, 30 and 35, each with shading to the right. A 32-year-old is a solution of $a \ge 25$ and of $a \ge 30$, but not of $a \ge 35$, so she could serve in the House or the Senate but could not yet run for president.
Pure water freezes at 32 degrees Fahrenheit. At temperatures below that, a puddle turns to ice. With $t$ for the air temperature in degrees Fahrenheit, 'freezing weather' is roughly $t < 32$, an open circle at 32 and shading to the left. Weather forecasters issue freeze warnings for gardeners when the temperature is expected to fall to 32 degrees or lower, which is $t \le 32$: now the circle is filled, because 32 itself is included.
Test some values. A winter morning at 20 degrees: $20 \le 32$ is true, so plants are in danger. An afternoon at 45 degrees: $45 \le 32$ is false, so they are safe. A reading of $-4$ degrees in Minnesota: $-4 \le 32$ is true, since every negative temperature is to the left of 32. One small line under the symbol decides whether the boundary day of exactly 32 degrees counts.
Thinking an inequality has one answer. $x > 3$ is not 'the answer is 4'. Every number greater than 3 works, including fractions and decimals.
Mixing up the circles. Open means the boundary is left out ($<$, $>$); filled means it is included ($\le$, $\ge$).
Shading the wrong way. Greater is to the right, less is to the left. Check with a test value.
Flipping 'at least'. At least 12 is $\ge 12$, not $\le 12$. Ask: is 13 allowed?
Reading the symbol backward when the number is first. $5 > x$ means $x$ is less than 5, so it is the same as $x < 5$. Put the letter first when you read it.
Forgetting 0 when counting. The whole numbers less than 6 are 0 through 5: six numbers, not five.
Graph $x \le 4$. Read the symbol aloud.
$x \text{ is less than or equal to } 4$
Saying it in words shows the solutions are 4 and the numbers below it.
Find the boundary.
$\text{boundary} = 4$
This is where the solutions stop.
Choose the circle.
$4 \le 4 \text{ is true, so a filled circle at } 4$
The line under the symbol lets the boundary be a solution.
Choose the direction.
$\text{less than} \;\Rightarrow\; \text{shade left}$
Smaller numbers lie to the left on a number line.
Check with a test value in the shading.
$0 \le 4 \text{ is true}$
A value on the shaded side must be a solution, and it is.
Overnight, the temperature stayed above $-3$ degrees Fahrenheit. Name the variable.
$t = \text{temperature in degrees Fahrenheit}$
The temperature is the quantity that the rule describes.
Translate 'above $-3$'.
$t > -3$
Above means greater than, and the boundary itself is not above.
Choose the circle at $-3$.
$-3 > -3 \text{ is false, so an open circle}$
A temperature of exactly $-3$ is not above $-3$.
Choose the direction.
$\text{greater than} \;\Rightarrow\; \text{shade right}$
Warmer temperatures are the greater numbers, to the right.
Test a value on the shaded side.
$1 > -3 \text{ is true}$
1 degree is warmer than $-3$ degrees, so it is a solution.
Test a value on the unshaded side.
$-5 > -3 \text{ is false}$
$-5$ is further left, colder, so it is correctly left unshaded.
A school bus has 48 seats and 30 students are already sitting down. How many more students can board? Name the variable.
$s = \text{number of students who can still board}$
The unknown is how many more students fit.
Find the seats still empty.
$48 - 30 = 18$
Only the empty seats are available.
Write the inequality.
$s \le 18$
At most 18 more can board: 18 is allowed, 19 is not.
Decide which numbers make sense.
$s = 0, 1, 2, \ldots, 18$
Students come in whole numbers, and 0 is possible if nobody else boards.
Count the possible answers.
$18 + 1 = 19$
A list from 0 to 18 has 19 members, because 0 counts too.
Check the boundary.
$30 + 18 = 48 \le 48$
With 18 more, every seat is full, which is allowed.
Check one past the boundary.
$30 + 19 = 49 > 48$
One more student would have no seat, so 19 is not a solution.
Translate 'at most 5'.
$n \le 5$
At most means that number or less.
Choose the circle.
$\text{filled circle at } 5$
5 itself is allowed, because the symbol includes 'equal to'.
Choose the direction of the shading.
Which picture on a number line shows the solutions of $x \ge 6$?
Which whole numbers make $x + 4 > 12$ true? Complete the solution by testing two values, then finding the boundary and the smallest whole-number solution.
Test $x = 7$.
$7 + 4 =$ low
This sum is less than 12, so $x = 7$ is not a solution.
Test $x = 10$.
$10 + 4 =$ high
This sum is greater than 12, so $x = 10$ is a solution.
Find the boundary: the value of $x$ that makes the two sides equal.
$x = 12 - 4 =$ edge
Subtraction undoes adding 4.
Decide whether the boundary is a solution.
$\text{equal is not greater, so an open circle}$
The symbol is $>$, with no 'or equal to' line, so the boundary is left out.
Find the smallest whole number that works.
$x =$ first
It is the first whole number to the right of the open circle; every whole number after it works too.
Match each rule to the inequality that says the same thing.
| $p \ge 68$ | $t < 5$ | $n > 159$ | $m \le 59$ | |
|---|---|---|---|---|
| You need at least 68 points to pass. | ||||
| Fewer than 5 tickets are left. | ||||
| More than 159 people came to the game. | ||||
| No more than 59 minutes of screen time. |
Is $x = 9$ a solution of $x > 9$?
How many whole numbers make $x < 14$ true? (The whole numbers are 0, 1, 2, 3 and so on.)
There are answer whole-number solutions.
To ride a roller coaster at an amusement park, you must be at least 38 inches tall. Give the set of heights $h$, in inches, of the people who are allowed to ride.
This task has no paper form; do it on a device.
A freight elevator can safely carry no more than 384 pounds. The operator weighs 146 pounds and rides along. Each box weighs 33 pounds. What is the greatest number of boxes the operator can take in one trip?
The operator can take at most answer boxes.
Lesson test: one question per skill, one attempt each, no hints. Your answers are checked when you submit.
To ride a roller coaster at an amusement park, you must be at least 52 inches tall. Give the set of heights $h$, in inches, of the people who are allowed to ride.
This task has no paper form; do it on a device.
You can write and graph an inequality. Without looking: what does an open circle mean, and how many solutions does $x > 3$ have?
16. Your turn: write and graph 'at most 5' for a number $n$, step 3
$\text{shade left from } 5$
Less than means smaller numbers, which are to the left.