Open hole vs closed hole math

You have already worked with inequality statements. Let's refresh those skills.

Índice Show

What is the difference between an open and closed circle in math?
What does an open dot mean in math?
When graphing do you use an open or closed circle for?

Inequality Notations: (see other notation forms at Notations for Solutions)
a > b ; a is strictly greater than b	a b ; a is greater than or equal to b
a < b ; a is striclty less than b	a b ; a is less than or equal to b
a ≠ b ; a is not equal to b	Hint: The "open" (larger) part of the inequality symbol always faces the larger quantity.

If you can solve a linear equation, you can solve a linear inequality. The process is the same, with one exception ...

... when you multiply (or divide) an inequality by a negative value,
you must change the direction of the inequality.

Let's see why this "exception" is actually needed.

We know that 3 is less than 7.
Now, lets multiply both sides by -1.
Examine the results (the products).

... written 3 < 7.
... written (-1)(3) ? (-1)(7)
... written -3 ? -7

On a number line, -3 is to the right of -7, making -3 greater than -7.

-3 > -7
We have to reverse the direction of the inequality, when we multiply by a negative value, in order to maintain a "true" statement.

When graphing a linear inequality on a number line, use an open circle for "less than" or "greater than", and a closed circle for "less than or equal to" or "greater than or equal to".

Graph the solution set of: -3 < x < 4

The solution set for this problem will be all values that satisfy both -3 < x and x < 4.

Look for where the two inequalities overlap.

Graph using open circles for -3 and 4 (since x can not equal -3 nor 4), and a bar to show the overlapping section.

Graph the solution set of: x < -3 or x

The solution set for this problem will be the full graph of both inequalities, since the two inequalities do not overlap.

Notice that there is one open circle (for -3) and one closed circle (for 1).

Solve and graph the solution set of: 4x < 24

Proceed as you would when solving a linear equation:
Divide both sides by 4.

Note: The direction of the inequality stays the same since we did NOT divide by a negative value.

Graph using an open circle for 6 (since x can not equal 6) and an arrow to the left (since our symbol is less than).

Solve and graph the solution set of: -5x

Divide both sides by -5.

Note: The direction of the inequality was reversed since we divided by a negative value.

Graph using a closed circle for -5 (since x can equal -5) and an arrow to the left (since our final symbol is less than or equal to).

Solve and graph the solution set of: 3x + 4 > 13

Proceed as you would when solving a linear equation:
Subtract 4 from both sides.
Divide both sides by 3.

Note: The direction of the inequality stays the same since we did NOT divide by a negative value.

Graph using an open circle for 3 (since x can not equal 3) and an arrow to the right (since our symbol is greater than).

Solve and graph the solution set of: 9 - 2x

Subtract 9 from both sides.
Divide both sides by -2.

Note: The direction of the inequality was reversed since we divided by a negative value.

Graph using a closed circle for 3 (since x can equal 3) and an arrow to the right (since our symbol is greater than or equal to).

What is the difference between an open and closed circle in math?

Open circles are used for numbers that are less than or greater than (< or >). Closed circles are used for numbers that are less than or equal to and greater than or equal to (≤ or ≥).

What does an open dot mean in math?

A solid dot on a number line graph indicates that the given number should be included as a possible solution, whereas an open dot indicates that the given number cannot be a solution. For example, if you graph x > 7, you place an open dot at 7 because it's not a valid answer (7 is not greater than itself).

When graphing do you use an open or closed circle for?

When graphing a linear inequality on a number line, use an open circle for "less than" or "greater than", and a closed circle for "less than or equal to" or "greater than or equal to".