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Interpreting
Inequality Solutions

Learn to read, write, and visualize solution sets of inequalities: what it means when \( x \geq 5 \), why there are infinitely many answers, and how to apply them in real-world contexts.

Pre-Algebra

What Is an Inequality?

An inequality is a mathematical statement that compares two expressions using one of these symbols: \( < \), \( > \), \( \leq \), or \( \geq \). Unlike an equation, which has one answer, an inequality usually has many possible answers.

\( x > 3 \)
"x is greater than 3": any number larger than 3 works.
\( x \leq 10 \)
"x is less than or equal to 10": 10 itself and anything smaller works.

The Four Inequality Symbols

\( < \) Less Than
Does not include the boundary value. Open circle on a number line.
\( > \) Greater Than
Does not include the boundary value. Open circle on a number line.
\( \leq \) Less Than or Equal
Includes the boundary value. Closed (filled) circle on a number line.
\( \geq \) Greater Than or Equal
Includes the boundary value. Closed (filled) circle on a number line.

What Is a Solution Set?

The solution set of an inequality is the collection of all values that make the inequality true. We can show it three ways: in words, with inequality notation, and on a number line.

0 1 2 3 4
\( x > 2 \)

Open vs Closed Circles

\( x > 3 \) (open circle)
 1 2 3 4 5
\( x \geq 3 \) (closed circle)
 1 2 3 4 5
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.

The Core Rule for Solutions

1
Identify the boundary. The number in the inequality (e.g., 5 in \( x > 5 \)) is your boundary.
2
Determine the direction. Is the solution set going left (smaller values) or right (larger values)?
3
Check the boundary. Is the boundary itself included (\( \leq \) or \( \geq \)) or excluded (\( < \) or \( > \))?
5
\( x < 5 \) or \( x \leq 5 \)
\( x > 5 \) or \( x \geq 5 \)

Inequality vs Equation

Feature Equation Inequality
Symbol \( = \) \( <, >, \leq, \geq \)
Number of solutions One (usually) Infinitely many
Graph on number line A single point A ray (shaded region)
Example \( x = 7 \) \( x \geq 7 \)
Boundary included? Always Depends on symbol

Inequalities in Real Life

Inequalities describe constraints and limits in everyday life. When we interpret a solution set in context, we explain what the range of values actually means in the real situation.

Speed Limits
\( s \leq 65 \): your speed must be at most 65 mph.
Height Requirements
\( h \geq 48 \): you must be at least 48 inches tall to ride.
Budget Limits
\( c < 20 \): the cost must be less than $20.

How to Interpret a Solution Set

1
Read the inequality. Identify the variable, the boundary value, and the direction of the symbol.
2
Translate to words. Write what the inequality says in plain English: "x is at least 12" or "t is less than 5."
3
Connect to context. Replace the variable with what it represents: "The temperature is at least 12 degrees."
4
Test a value. Pick a number from the solution set and verify it makes sense in the scenario.
5
Consider reasonableness. In context, the range might have practical limits (e.g., no negative heights).

Example 1: Amusement Park Ride

A roller coaster requires riders to be at least 48 inches tall. Let \( h \) represent height in inches. The inequality is \( h \geq 48 \).

42 44 46 48 50
\( h \geq 48 \)

Example 2: Budget Constraint

Maya has $25 to spend on books. Each book costs $8. She wants to know how many books \( b \) she can buy. The inequality is \( 8b \leq 25 \), which simplifies to \( b \leq 3.125 \).

0 1 2 3
\( b \leq 3.125 \)

Example 3: Temperature Range

A science experiment requires the temperature to stay below \( 100^{\circ}\text{F} \). Let \( t \) represent the temperature. The inequality is \( t < 100 \).

96 97 98 99 100
\( t < 100 \)

Considering Practical Limits

A solution set on a number line goes on forever, but real-world context adds natural boundaries that narrow the range of reasonable values.

No Negatives
Quantities like height, weight, and number of items can't be negative.
Whole Numbers Only
You can't buy 2.7 tickets or invite 4.5 friends: some variables must be integers.
Upper Bounds
A class has at most 30 students; a day has at most 24 hours: context sets a maximum.

Key Phrases and Their Symbols

English Phrase Symbol Boundary Included?
"at least" \( \geq \) Yes (closed)
"at most" \( \leq \) Yes (closed)
"more than" \( > \) No (open)
"fewer than" / "less than" \( < \) No (open)
"no more than" \( \leq \) Yes (closed)
"no fewer than" \( \geq \) Yes (closed)

Common Mistakes to Avoid

✕ Open vs Closed Mix-Up
Using an open circle for \( \leq \) or \( \geq \), or a closed circle for \( < \) or \( > \). Remember: the line under the symbol means closed.
✕ Ignoring Context
Saying "all numbers greater than 2" when the problem asks about whole books or number of people. Always filter the solution set through the context.
✕ Wrong Direction
Shading the number line to the right for \( x < 5 \) instead of to the left. The arrow must point toward the values that satisfy the inequality.

Question 1 of 5

A sign at a pool says "You must be at least 10 years old to swim without an adult." Which inequality represents the ages of swimmers who can swim alone?

Question 2 of 5

The solution to an inequality is graphed with an open circle at 7 and shading to the left. Which inequality does this represent?

Question 3 of 5

A student solves \( 3n \leq 18 \) and gets \( n \leq 6 \). If \( n \) represents the number of friends invited to a party, which values of \( n \) are reasonable?

Question 4 of 5

Carlos has more than $50 in his savings account. Which number line correctly shows the solution set?

A 50 B 50 C 50 D 50

Question 5 of 5

A school bus can carry no more than 48 passengers. Let \( p \) be the number of passengers. Which statement best interprets the solution set of \( p \leq 48 \)?