01 / 20

Checking
Boundary Values

Learn how to test whether endpoints belong in a solution set and justify your reasoning with open and closed circles.

Pre-Algebra

What Is a Boundary Value?

A boundary value is the number right at the edge of an inequality's solution set. It is the value where the inequality "switches" from true to false (or false to true).

3 boundary solution region not solutions 1 2 4 5

Open vs Closed Circles

Open Circle ○
Used for \( < \) and \( > \). The boundary value is not included in the solution set.
Closed Circle ●
Used for \( \leq \) and \( \geq \). The boundary value is included in the solution set.
5 \( x > 5 \) 5 \( x \geq 5 \)

Why Boundaries Matter

A boundary value is the tipping point of an inequality. Getting it wrong means your solution set is either missing a valid answer or including one that does not work.

Real-Life Example
A roller coaster requires height \( \geq 48 \) inches. A child who is exactly 48 inches can ride: the boundary is included.
Math Example
If \( x > 7 \), then \( x = 7 \) does not satisfy the inequality, even though it is right at the edge.

The Substitution Test

To verify whether a boundary value belongs in the solution set, follow this rule:

1
Solve the inequality to find the boundary value.
2
Plug that value back into the original inequality.
3
Evaluate: is the statement true or false?
4
Conclude: true → closed circle ●  |  false → open circle ○

Matching Symbols to Circles

Symbol Meaning Circle \( < \) less than \( > \) greater than \( \leq \) less than or equal to \( \geq \) greater than or equal to open = excluded closed = included

\( \leq \) vs \( < \): The Critical Difference

\( x \leq 4 \) \( x < 4 \)
Boundary value 4 4
Is \( x = 4 \) a solution? Yes (\( 4 \leq 4 \) is true) No (\( 4 < 4 \) is false)
Circle type Closed ● Open ○
Is \( x = 3 \) a solution? Yes Yes
Is \( x = 5 \) a solution? No No

How to Check a Boundary

1
Isolate the variable to find the boundary. For \( 2x + 1 > 9 \), solve to get \( x > 4 \). The boundary is 4.
2
Look at the inequality symbol. Is it strict (\( <, > \)) or non-strict (\( \leq, \geq \))?
3
Substitute the boundary value into the original inequality: \( 2(4) + 1 > 9 \) becomes \( 9 > 9 \).
4
Evaluate: \( 9 > 9 \) is false, so \( x = 4 \) is not in the solution set. Use an open circle.
5
Justify: Write "Since \( 9 > 9 \) is false, the endpoint is excluded."

Example 1: \( x \geq 3 \)

Question: Does \( x = 3 \) belong in the solution set of \( x \geq 3 \)?

1
The boundary value is 3.
2
Substitute: \( 3 \geq 3 \).
3
Evaluate: \( 3 \geq 3 \) is true (3 is equal to 3).
1 2 3 4 5 included

Example 2: \( 3x - 5 < 10 \)

Question: Solve and determine whether the boundary value is included.

1
Solve: \( 3x - 5 < 10 \) → \( 3x < 15 \) → \( x < 5 \). Boundary is 5.
2
Substitute \( x = 5 \) into the original: \( 3(5) - 5 < 10 \) → \( 10 < 10 \).
3
Evaluate: \( 10 < 10 \) is false. So \( x = 5 \) is excluded.
3 4 5 6 7 excluded

Example 3: \( -2x + 8 \geq 2 \)

Question: Solve, check the boundary, and graph on a number line.

1
Solve: \( -2x + 8 \geq 2 \) → \( -2x \geq -6 \) → \( x \leq 3 \) (flip the sign when dividing by a negative).
2
Substitute \( x = 3 \): \( -2(3) + 8 \geq 2 \) → \( 2 \geq 2 \).
3
Evaluate: \( 2 \geq 2 \) is true. Boundary 3 is included.
1 2 3 4 5 included

Example 4: Two Boundaries

Question: For \( 1 < x \leq 6 \), check both boundary values.

Left Boundary: \( x = 1 \)
Check \( 1 < 1 \): this is false. So \( x = 1 \) is excluded (open circle).
Right Boundary: \( x = 6 \)
Check \( 6 \leq 6 \): this is true. So \( x = 6 \) is included (closed circle).
0 1 2 3 4 5 6 7 open closed

Common Mistakes

✕ Confusing the Symbols
Using a closed circle for \( < \) or \( > \). Remember: no "equal to" means no fill.
✕ Forgetting to Flip
When dividing by a negative, the inequality flips. Then the boundary check must use the original inequality.
✕ Checking the Wrong Equation
Substituting into the simplified form instead of the original inequality can hide errors from earlier steps.

Summary Reference Table

Inequality Circle Boundary In Set? Justification
\( x > a \) Open ○ No \( a > a \) is false
\( x < a \) Open ○ No \( a < a \) is false
\( x \geq a \) Closed ● Yes \( a \geq a \) is true
\( x \leq a \) Closed ● Yes \( a \leq a \) is true

Ready to Practice?

Let's review the key steps one more time before the assessment questions:

1
Find the boundary value by solving the inequality.
2
Substitute it back into the original inequality.
3
Decide: true → closed circle (included)  |  false → open circle (excluded).
4
Justify your answer by showing the substitution result.

Question 1 of 5

For the inequality \( x \leq 7 \), is the boundary value \( x = 7 \) included in the solution set?

Question 2 of 5

Solve \( 4x + 2 > 18 \). What type of circle goes at the boundary value?

Question 3 of 5

For \( 2 \leq x < 9 \), which statement is correct about the boundary values?

Question 4 of 5

A student solves \( -3x + 6 \leq 0 \) and gets \( x \leq -2 \). They test \( x = -2 \) in the original and get \( 12 \leq 0 \), which is false. What went wrong?

Question 5 of 5

Which justification correctly explains why \( x = 5 \) is not in the solution set of \( x > 5 \)?