Solved Exercises on Graphing Inequalities in Grade 7

Master graphing inequalities: number lines, greater than, less than, compound inequalities, and real-world applications through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Less Than Inequality
Exercise 1
Graph: x < 5
Definition:

Graphing inequality: Representing all solutions to an inequality on a number line.

Graphing method:
  1. Locate the boundary point on the number line
  2. Determine if the boundary is included (closed dot) or excluded (open dot)
  3. Shade in the direction that satisfies the inequality
Inequality
x < 5
Boundary Point
5
Dot Type
Open dot (not included)
Step 1: Locate boundary point

Place a point at 5 on the number line

Step 2: Determine dot type

Since x < 5 (not x ≤ 5), 5 is not included

Use an open circle at 5

Step 3: Shade direction

Since x is less than 5, shade to the left of 5

Step 4: Verify

Test a value in the shaded region: x = 3 → 3 < 5 ✓

Test the boundary: x = 5 → 5 < 5 ✗

Open circle at 5, shading to the left
Final answer:

Open circle at 5 with shading to the left

Applied rules:

• Less than: Open dot, shade to the left

• Greater than: Open dot, shade to the right

• Inclusive: Closed dot (≤ or ≥)

• Exclusive: Open dot (< or >)

2 Greater Than or Equal Inequality
Exercise 2
Graph: x ≥ -2
Definition:

Greater than or equal: Uses a closed dot since the boundary value is included in the solution.

Inequality
x ≥ -2
Boundary Point
-2
Dot Type
Closed dot (included)
Step 1: Locate boundary point

Place a point at -2 on the number line

Step 2: Determine dot type

Since x ≥ -2 (includes -2), use a closed circle at -2

Step 3: Shade direction

Since x is greater than or equal to -2, shade to the right of -2

Step 4: Verify

Test a value in the shaded region: x = 0 → 0 ≥ -2 ✓

Test the boundary: x = -2 → -2 ≥ -2 ✓

Closed circle at -2, shading to the right
Final answer:

Closed circle at -2 with shading to the right

Applied rules:

• Greater than or equal: Closed dot, shade to the right

• Less than or equal: Closed dot, shade to the left

• Inclusive inequalities: Use closed dots (≥ or ≤)

3 Compound Inequality
Exercise 3
Graph: -3 < x ≤ 4
Definition:

Compound inequality: Two inequalities joined by "and" or "or", representing a range of values.

Compound Inequality
-3 < x ≤ 4
Left Boundary
-3 (open dot)
Right Boundary
4 (closed dot)
Step 1: Identify boundaries

Left boundary: -3 (exclusive due to <)

Right boundary: 4 (inclusive due to ≤)

Step 2: Draw boundary points

Open circle at -3 (not included)

Closed circle at 4 (included)

Step 3: Shade between boundaries

Shade the region between -3 and 4

Step 4: Verify

Test a value in the shaded region: x = 0 → -3 < 0 ≤ 4 ✓

Test boundary values: x = -3 → -3 < -3 ≤ 4 ✗, x = 4 → -3 < 4 ≤ 4 ✓

Open circle at -3, closed circle at 4, shading between
Final answer:

Open circle at -3, closed circle at 4, with shading between them

Applied rules:

• Compound "and" inequalities: Shaded region between boundaries

• Dot types: Open for < and >, closed for ≤ and ≥

• Verification: Test values within and at boundaries

Graphing Inequalities Rules and Methods
x > a \Rightarrow \text{open dot at } a, \text{ shade right}
Greater Than Inequality
Less Than
x < a → open dot at a, shade left
Excludes boundary value
Greater Than
x > a → open dot at a, shade right
Excludes boundary value
Inclusive
x ≤ a or x ≥ a → closed dot, includes boundary
Uses filled circle
Key definitions:

Inequality: A mathematical statement that compares two expressions using <, >, ≤, or ≥.

Boundary point: The value that separates the solution from the non-solution on the number line.

Open dot: Used for < and > inequalities, indicating the boundary value is not included.

Closed dot: Used for ≤ and ≥ inequalities, indicating the boundary value is included.

Compound inequality: Two or more inequalities joined by "and" or "or" to describe ranges of values.

Graphing methodology:
  1. Identify the boundary: Find the value that defines the inequality
  2. Determine dot type: Open for < or >, closed for ≤ or ≥
  3. Choose direction: Left for < or ≤, right for > or ≥
  4. Shade appropriately: Fill the region that satisfies the inequality
  5. Verify solution: Test values in the shaded and unshaded regions
Tip 1: Remember: open dots for < and >, closed dots for ≤ and ≥.
Tip 2: On number lines, smaller numbers are to the left, larger to the right.
Tip 3: Test a point in the shaded region to verify your graph is correct.
Tip 4: For compound inequalities, both conditions must be satisfied simultaneously.
Common errors: Using wrong dot type, shading in wrong direction, misreading inequality symbols, not considering whether boundary values are included.
Exam preparation: Practice all inequality types, master dot selection, understand compound inequalities, work with negative numbers and fractions.
Essential rules:

• < and >: Open dot, exclude boundary value

• ≤ and ≥: Closed dot, include boundary value

• < and ≤: Shade to the left

• > and ≥: Shade to the right

• Verification: Test values in shaded and unshaded regions

Solution: Exercises 4 to 5
4 Two-Sided Inequality
Exercise 4
Graph: x ≤ -1 or x > 3
Definition:

"Or" compound inequality: Solutions satisfy either condition, resulting in two separate shaded regions.

Compound Inequality
x ≤ -1 or x > 3
Left Region
x ≤ -1 (closed dot at -1, shade left)
Right Region
x > 3 (open dot at 3, shade right)
Step 1: Analyze each part separately

x ≤ -1: Includes -1, shades to the left

x > 3: Excludes 3, shades to the right

Step 2: Draw left region

Closed circle at -1, shade everything to the left

Step 3: Draw right region

Open circle at 3, shade everything to the right

Step 4: Verify

Test x = -2: -2 ≤ -1 ✓ or -2 > 3 ✗ → Included

Test x = 4: 4 ≤ -1 ✗ or 4 > 3 ✓ → Included

Test x = 0: 0 ≤ -1 ✗ or 0 > 3 ✗ → Not included

Closed circle at -1 with left shading, open circle at 3 with right shading
Final answer:

Two separate regions: closed dot at -1 with left shading, open dot at 3 with right shading

Applied rules:

• "Or" inequalities: At least one condition must be satisfied

• Disjoint regions: "Or" can create multiple separate shaded areas

• Verification: Test values in each region to confirm

5 Real-World Application
Exercise 5
A store offers free shipping for orders over $25. Graph the inequality representing order amounts that qualify for free shipping.
Definition:

Real-world applications: Translating practical situations into inequalities and graphing solutions.

Problem Setup
Let x = order amount
Write Inequality
x > 25
Graph
Open dot at 25, shade right
Step 1: Define the variable

Let x = the order amount in dollars

Step 2: Set up the inequality

Free shipping for orders over $25 means x > 25

Step 3: Graph the inequality

Open circle at 25 (since $25 is not over $25)

Shade to the right (values greater than 25)

Step 4: Interpret the solution

Any order amount greater than $25 qualifies for free shipping

Step 5: Verify with examples

$30 > 25 ✓ (qualifies for free shipping)

$20 > 25 ✗ (does not qualify)

Open circle at 25, shading to the right
Final answer:

Open circle at 25 with shading to the right represents all qualifying order amounts

Applied rules:

• Problem translation: Convert real-world conditions to mathematical inequalities

• Context consideration: Understand whether boundary values are included

• Graph interpretation: Relate the mathematical solution back to the real-world situation

Detailed Graphing Inequalities Guide
x > a \Rightarrow \text{open dot at } a, \text{ shade right}
Inequality Graphing Rule
Key definitions:

Inequality: A mathematical statement that compares two expressions using inequality symbols (<, >, ≤, ≥). It represents a range of possible values rather than a single solution.

Graphing inequalities: The process of representing all possible solutions to an inequality on a number line, showing the range of values that satisfy the condition.

Boundary point: The critical value that separates the solution region from the non-solution region on the number line.

Open dot: A hollow circle used on the number line to indicate that the boundary value is NOT included in the solution set (< or >).

Closed dot: A filled circle used on the number line to indicate that the boundary value IS included in the solution set (≤ or ≥).

Compound inequality: Two or more inequalities combined with "and" or "or" to describe more complex solution sets.

Complete graphing methodology:
  1. Identify the inequality: Determine the comparison symbol and boundary value
  2. Locate the boundary point: Mark the boundary value on the number line
  3. Choose dot type: Open for < or >, closed for ≤ or ≥
  4. Determine shading direction: Left for < or ≤, right for > or ≥
  5. Draw the graph: Place the dot and shade the appropriate region
  6. Verify the solution: Test values in both shaded and unshaded regions
Tip 1: Remember "OIL" - Open dot for Inequality with Less than (<) or Greater than (>).
Tip 2: On the number line, values increase as you move right and decrease as you move left.
Tip 3: For "and" compound inequalities, the solution is the intersection (overlap) of both conditions.
Tip 4: For "or" compound inequalities, the solution includes all values satisfying either condition.
Common errors: Confusing dot types, shading in wrong direction, misreading inequality symbols, not considering whether boundary values are included, misunderstanding compound inequalities with "and" vs "or".
Applications: Budget constraints, age requirements, temperature ranges, speed limits, measurement tolerances, business thresholds, and representing solution sets for equations and functions.
Essential graphing rules:

• < and >: Use open dot, exclude boundary value

• ≤ and ≥: Use closed dot, include boundary value

• < and ≤: Shade to the left of boundary

• > and ≥: Shade to the right of boundary

• "And" inequalities: Intersection of both conditions

• "Or" inequalities: Union of both conditions

Graphing Inequalities Guide

📊
Less Than

x < 3

→ Open dot at 3

→ Shade left

Greater Than

x > 3

→ Open dot at 3

→ Shade right

Inclusive

x ≤ 3 or x ≥ 3

→ Closed dot

→ Include boundary

Verification

Test values in

shaded regions

to confirm

Questions & Answers

Question: How do I know whether to use an open dot or closed dot when graphing inequalities?

Answer: The dot type depends on whether the boundary value is included in the solution:

Open dot (○): Use for strict inequalities

  • < (less than) - boundary is NOT included
  • > (greater than) - boundary is NOT included

Closed dot (●): Use for inclusive inequalities

  • ≤ (less than or equal to) - boundary IS included
  • ≥ (greater than or equal to) - boundary IS included

Memory trick: Think of the inequality symbol:

  • < and > have no "equal" line, so boundary is not included (open dot)
  • ≤ and ≥ have an "equal" line, so boundary is included (closed dot)

Example: For x < 5, use open dot at 5 because 5 is not less than 5.

Question: When I graph x ≥ 4, should I shade to the left or right? How do I remember?

Answer: For x ≥ 4, shade to the RIGHT. Here's how to remember:

Number line orientation:

  • Numbers increase as you move right
  • Numbers decrease as you move left

For x ≥ 4:

  • x must be greater than or equal to 4
  • Values like 5, 6, 7... are greater than 4
  • These values are located to the RIGHT of 4 on the number line

Memory aids:

  • < (less than) → shade LEFT (smaller values)
  • > (greater than) → shade RIGHT (larger values)
  • Think "less" = "left", "great" = "right"

So for x ≥ 4: closed dot at 4, shading to the right.

Question: What does it mean when I have "and" or "or" in a compound inequality like x > 2 and x < 8?

Answer: "And" and "or" connect conditions differently:

"And" inequalities: Both conditions must be true simultaneously

  • x > 2 and x < 8 means x must satisfy BOTH conditions
  • x > 2: x is greater than 2
  • x < 8: x is less than 8
  • Combined: 2 < x < 8 (values between 2 and 8)
  • Graph: Shading BETWEEN the two boundary points

"Or" inequalities: Either condition can be true

  • x > 5 or x < 1 means x satisfies AT LEAST ONE condition
  • x > 5: values greater than 5
  • x < 1: values less than 1
  • Combined: All values except those between 1 and 5
  • Graph: Shading OUTSIDE the boundary points

Visual: "And" creates overlap (intersection), "or" creates union (both regions).