Graphing x < 4

Graph the inequality $x < 4$ on a number line.

Identify the boundary point: The number after the symbol is 4 = Boundary: 4

Determine the circle type: The symbol is $<$ (strict inequality, not equal) = Open circle ○

Decide shading direction: $x < 4$ means all numbers LESS than 4 = Shade LEFT (toward smaller numbers)

Draw the graph: Draw open circle at 4, shade arrow pointing left = ○←———— at 4

Answer: Open circle at 4, arrow pointing left. Solutions include 3, 2, 1, 0, -1, and all numbers less than 4.

Graphing x ≥ -2

Graph the inequality $x \geq -2$ on a number line.

Identify the boundary point: The number is $-2$ = Boundary: $-2$

Determine the circle type: The symbol is $\geq$ (includes equal to) = Closed circle ●

Decide shading direction: $x \geq -2$ means all numbers GREATER than or equal to $-2$ = Shade RIGHT (toward larger numbers)

Draw the graph: Draw closed circle at $-2$, shade arrow pointing right = ●————→ at $-2$

Answer: Closed circle at $-2$, arrow pointing right. Solutions include $-2$, $-1$, 0, 1, 2, and all numbers greater than $-2$.

Writing an Inequality from a Graph

A number line shows a closed circle at 5 with shading to the left. Write the inequality.

Identify the boundary: The circle is at 5 = Boundary: 5

Interpret the circle type: Closed circle means the boundary IS included = Use $\leq$ or $\geq$

Interpret the shading: Shading goes LEFT, toward smaller numbers = Values are LESS than or equal to 5

Write the inequality: Combine: $x$ is less than or equal to 5 = $x \leq 5$

Answer: $x \leq 5$

Solving and Graphing

Solve $x + 3 > 7$ and graph the solution.

Isolate x: $x + 3 > 7$ $x + 3 - 3 > 7 - 3$ = $x > 4$

Identify the boundary: The simplified inequality shows 4 = Boundary: 4

Determine circle type: $>$ means strict inequality (not equal) = Open circle ○

Draw the graph: Open circle at 4, shade right (greater than) = ○————→ at 4

Answer: $x > 4$: Open circle at 4, arrow pointing right.

Graphing with Negative Boundary

Graph $x \leq -3$ on a number line.

Identify the boundary point: The number is $-3$ = Boundary: $-3$

Determine the circle type: $\leq$ includes the equal sign = Closed circle ●

Decide shading direction: $x \leq -3$ means numbers less than or equal to $-3$ = Shade LEFT

Check with test values: Is $-4 \leq -3$? Yes! Is $-2 \leq -3$? No! = Shading left is correct

Answer: Closed circle at $-3$, arrow pointing left. Includes $-3$, $-4$, $-5$, etc.

Mistake: Using a closed circle for $<$ or $>$

Why: Students forget that $<$ and $>$ are strict inequalities that do NOT include the boundary point.

Correct: Use open circle for $<$ and $>$ (not included). Use closed circle for $\leq$ and $\geq$ (included).

Mistake: Shading the wrong direction

Why: Confusion about which way is 'greater' or 'less' on the number line.

Correct: Less than ($<$, $\leq$) → shade LEFT. Greater than ($>$, $\geq$) → shade RIGHT. Remember: right is greater!

Mistake: Confusing $-5 < -3$ with $-5 > -3$

Why: With negative numbers, the digit size can be misleading.

Correct: On a number line, $-5$ is to the LEFT of $-3$, so $-5 < -3$. Further left = smaller.

Mistake: Forgetting to flip the inequality when multiplying by negative

Why: When solving inequalities, multiplying or dividing by a negative reverses the direction.

Correct: If you multiply or divide by a negative number, flip the inequality symbol!

Theme Park Height Requirements

Theme parks use inequalities to set minimum heights for rides.

A roller coaster requires riders to be at least 140 cm tall. This is written as $h \geq 140$ and graphed with a closed circle at 140, shading right.

Temperature Safety for Food

Food safety guidelines use inequalities to show safe storage temperatures.

Refrigerated food should be kept below 5°C. This is $t < 5$: open circle at 5, shade left.

Speed Limits

Traffic laws use inequalities to define legal speeds.

In a school zone, speed must be less than 30 km/h. This is $s < 30$: open circle at 30, shade left.

Q: How do I remember which circle to use?

A: Think of the line under $\leq$ and $\geq$ as 'filling in' the circle. No line = open circle. Line underneath = closed (filled) circle.

Q: What if the inequality is written 'backwards' like $5 > x$?

A: Rewrite it with $x$ first: $5 > x$ is the same as $x < 5$. The number that $x$ is compared to (5) is your boundary point.

Q: Can I check if my graph is correct?

A: Yes! Pick a number from your shaded region and substitute it into the original inequality. If it makes the inequality true, your graph is correct.