Solved Exercises on Graphing Linear Inequalities in Two Variables in Integrated Math 1

Master graphing linear inequalities: boundary lines, shading regions, and solution sets through these 5 detailed exercises with comprehensive solutions.

Solution: Exercises 1 to 3
1 Basic Graphing
Exercise 1
Graph the inequality \(y > 2x - 3\). Shade the solution region and identify the boundary line type.
Definition:

Linear Inequality: A mathematical statement involving linear expressions connected by inequality symbols (<, >, ≤, ≥)

Method for graphing linear inequalities:
  1. Graph the boundary line (solid for ≤ or ≥, dashed for < or >)
  2. Choose a test point not on the boundary line (usually origin if not on line)
  3. Substitute the test point into the inequality
  4. If true, shade the side containing the test point; if false, shade the opposite side
  5. Label the solution region
Inequality
y > 2x - 3
Boundary
y = 2x - 3 (dashed)
Shading
Above the line
Step 1: Identify the boundary line equation

The boundary line is \(y = 2x - 3\) (replace inequality with equals sign)

Step 2: Determine the line type

Since the inequality is \(y > 2x - 3\) (strict inequality), use a dashed line

Step 3: Graph the boundary line

Slope: 2, Y-intercept: -3

Points: (0, -3), (1, -1), (2, 1)

Step 4: Choose a test point

Use the origin (0, 0) since it's not on the boundary line

Step 5: Test the point in the inequality

Substitute (0, 0) into \(y > 2x - 3\):

\(0 > 2(0) - 3\)

\(0 > -3\)

This is true

Step 6: Shade the appropriate region

Since the test point (0, 0) satisfies the inequality, shade the side containing (0, 0)

This is the region above the boundary line

Step 7: Label the graph

Indicate the boundary line equation and the inequality symbol

Shaded region: y > 2x - 3
Final answer:

The graph shows the region above the dashed line \(y = 2x - 3\), representing all points where \(y > 2x - 3\).

Applied rules:

• Boundary Line: Solid for ≤, ≥; Dashed for <, >

• Test Point Method: Determine which side to shade

• Solution Region: All points satisfying the inequality

2 Less Than with Boundary
Exercise 2
Graph the inequality \(3x + 2y \leq 6\). Show the boundary line and shaded solution region.
Definition:

Standard Form Inequality: An inequality of the form \(ax + by \leq c\) or \(ax + by \geq c\)

Inequality
3x + 2y ≤ 6
Boundary
3x + 2y = 6 (solid)
Shading
Below the line
Step 1: Convert to slope-intercept form to graph

\(3x + 2y \leq 6\)

\(2y \leq -3x + 6\)

\(y \leq -\frac{3}{2}x + 3\)

Step 2: Identify the boundary line

Boundary line: \(y = -\frac{3}{2}x + 3\)

Since inequality is ≤, use a solid line

Step 3: Find intercepts to graph the line

X-intercept: Set \(y = 0\): \(3x = 6\), so \(x = 2\), point (2, 0)

Y-intercept: Set \(x = 0\): \(2y = 6\), so \(y = 3\), point (0, 3)

Step 4: Choose a test point

Use origin (0, 0): \(3(0) + 2(0) = 0 \leq 6\) ✓

Step 5: Shade the appropriate region

Since (0, 0) satisfies the inequality, shade the side containing (0, 0)

This is the region below the boundary line

Step 6: Verify with another point in the shaded region

Test point (1, 1): \(3(1) + 2(1) = 5 \leq 6\) ✓

Step 7: Draw the complete graph

Draw solid line through (2, 0) and (0, 3), shade below the line

Shaded region: 3x + 2y ≤ 6
Final answer:

The graph shows the region below and including the solid line \(3x + 2y = 6\), representing all points where \(3x + 2y \leq 6\).

Applied rules:

• Less Than/Equal: Solid line, shade toward test point if inequality is satisfied

• Standard Form Conversion: Solve for y to identify slope and y-intercept

• Intercept Method: Use x and y intercepts to graph the boundary line

3 Greater Than or Equal
Exercise 3
Graph the inequality \(y \geq -x + 4\). Identify the boundary line, shading direction, and a point that satisfies the inequality.
Definition:

Greater Than or Equal: Includes the boundary line in the solution set

Inequality
y ≥ -x + 4
Boundary
y = -x + 4 (solid)
Shading
Above the line
Step 1: Identify the boundary line

Boundary line: \(y = -x + 4\)

Slope: -1, Y-intercept: 4

Step 2: Determine the line type

Since the inequality is \(y \geq -x + 4\) (includes equals), use a solid line

Step 3: Graph the boundary line

Points: (0, 4), (1, 3), (4, 0)

Step 4: Choose a test point

Use origin (0, 0): \(0 \geq -0 + 4\) → \(0 \geq 4\) (false)

Step 5: Shade the appropriate region

Since (0, 0) does not satisfy the inequality, shade the opposite side

This is the region above the boundary line

Step 6: Identify a point that satisfies the inequality

Choose a point in the shaded region, such as (0, 5)

Check: \(5 \geq -0 + 4\) → \(5 \geq 4\) ✓

Step 7: Verify the boundary line is included

Points on the line satisfy the inequality (because of ≥)

Shaded region: y ≥ -x + 4
Final answer:

The graph shows the region above and including the solid line \(y = -x + 4\), representing all points where \(y \geq -x + 4\). A point that satisfies the inequality is (0, 5).

Applied rules:

• Greater Than or Equal: Solid line, shade away from test point if inequality is not satisfied

• Boundary Inclusion: Points on solid line are part of solution set

• Shading Direction: Above line for y ≥ mx + b

Graphing Linear Inequalities Rules and Methods
\(ax + by \, \square \, c\)
Linear Inequality in Two Variables
Strict Inequality
< or >
Dashed boundary line
Non-Strict
≤ or ≥
Solid boundary line
Shading Test
Test point method
Substitute and verify
Key definitions:

Linear Inequality in Two Variables: An inequality that can be written in the form \(ax + by \, \square \, c\) where a, b, and c are real numbers and a and b are not both zero

Boundary Line: The line formed by replacing the inequality symbol with an equals sign

Solution Region: The area of the coordinate plane containing all points that satisfy the inequality

Half-Plane: The region on one side of a line in the coordinate plane

Complete methodology:
  1. Identify Inequality Type: Determine if strict (<, >) or non-strict (≤, ≥)
  2. Graph Boundary Line: Draw as solid (≤, ≥) or dashed (<, >)
  3. Choose Test Point: Select a point not on the boundary line
  4. Substitute Test Point: Plug coordinates into original inequality
  5. Determine Shading: Shade appropriate half-plane based on test result
  6. Verify Solution: Check that a point in the shaded region satisfies the inequality
Tip 1: Always use the origin (0,0) as a test point if it's not on the boundary line.
Tip 2: For y > mx + b, shade above the line; for y < mx + b, shade below.
Tip 3: Solid lines include the boundary; dashed lines exclude it.
Tip 4: If the test point doesn't work, shade the opposite side.
Common errors: Confusing boundary line types, shading wrong side, forgetting to include boundary for ≤ and ≥.
Exam preparation: Practice different inequality forms, memorize boundary line rules, work with word problems.
Formulas to know by heart:

• Boundary line: Replace inequality with equals sign

• Line type: Strict (<, >) → dashed; Non-strict (≤, ≥) → solid

• Shading: Test point method determines which side to shade

Solution: Exercises 4 to 5
4 Word Problem Application
Exercise 4
A store sells pens for $2 each and notebooks for $3 each. A customer has at most $12 to spend. Write an inequality representing this situation and graph it. Identify at least one possible combination of pens and notebooks the customer could buy.
Definition:

Word Problem Modeling: Translating real-world constraints into mathematical inequalities

Variables
x=pens, y=notebooks
Inequality
2x + 3y ≤ 12
Solution
(3, 2)
Step 1: Define variables

Let \(x\) = number of pens

Let \(y\) = number of notebooks

Step 2: Identify the constraint

The customer has "at most" $12, meaning the total cost is less than or equal to $12

Step 3: Write the inequality

Total cost: \(2x + 3y \leq 12\)

Step 4: Graph the boundary line

Boundary line: \(2x + 3y = 12\)

Since inequality is ≤, use a solid line

X-intercept: (6, 0), Y-intercept: (0, 4)

Step 5: Choose and test a point

Use origin (0, 0): \(2(0) + 3(0) = 0 \leq 12\) ✓

Step 6: Shade the appropriate region

Since (0, 0) satisfies the inequality, shade the side containing the origin

Step 7: Identify a feasible solution

Any point in the shaded region represents a possible combination

Example: (3, 2) → 3 pens and 2 notebooks

Check: \(2(3) + 3(2) = 6 + 6 = 12 \leq 12\) ✓

2x + 3y ≤ 12, Solution: (3, 2)
Final answer:

The inequality is \(2x + 3y \leq 12\). The customer could buy 3 pens and 2 notebooks, which costs exactly $12.

Applied rules:

• Word Problem Translation: "at most" translates to ≤

• Constraint Modeling: Total cost ≤ budget

• Feasible Solutions: Points in shaded region satisfy constraints

5 Real-world Scenario
Exercise 5
A company produces two products: Product A requires 2 hours of labor and Product B requires 1 hour of labor. The company has at least 8 hours of labor available per day. Write an inequality and graph it. What does the solution region represent?
Definition:

Resource Constraint: A limitation on available resources that can be expressed as an inequality

Variables
x=A units, y=B units
Inequality
2x + y ≥ 8
Interpretation
Labor constraint
Step 1: Define variables

Let \(x\) = number of units of Product A

Let \(y\) = number of units of Product B

Step 2: Identify the constraint

The company has "at least" 8 hours of labor, meaning the total labor used must be greater than or equal to 8

Step 3: Write the inequality

Total labor used: \(2x + y \geq 8\)

Step 4: Graph the boundary line

Boundary line: \(2x + y = 8\)

Since inequality is ≥, use a solid line

X-intercept: (4, 0), Y-intercept: (0, 8)

Step 5: Choose and test a point

Use origin (0, 0): \(2(0) + 0 = 0 \geq 8\) (false)

Step 6: Shade the appropriate region

Since (0, 0) does not satisfy the inequality, shade the opposite side

This is the region above the boundary line

Step 7: Interpret the solution region

The shaded region represents all possible combinations of products A and B that require at least 8 hours of labor

Any point in this region is a feasible production plan

Step 8: Verify with an example

Test point (3, 3): \(2(3) + 3 = 9 \geq 8\) ✓

This means producing 3 units of A and 3 units of B uses 9 hours of labor, which meets the constraint

2x + y ≥ 8, Solution region: Above line
Final answer:

The inequality is \(2x + y \geq 8\). The solution region represents all combinations of products A and B that utilize at least 8 hours of labor per day.

Applied rules:

• Resource Constraints: "at least" translates to ≥

• Production Modeling: Total resource usage ≥ minimum requirement

• Feasible Region: Area representing all viable solutions

Linear Inequalities in Two Variables Fundamentals
\(ax + by \, \square \, c\)
Linear Inequality in Two Variables
Key definitions:

Linear Inequality: An inequality that can be written in the form \(ax + by \, \square \, c\) where a, b, and c are real numbers and a and b are not both zero

Boundary Line: The line formed by replacing the inequality symbol with an equals sign

Solution Set: The set of all ordered pairs (x, y) that make the inequality true

Half-Plane: The region of the coordinate plane on one side of a line

Complete methodology:
  1. Identify Inequality Symbol: Determine if strict (<, >) or non-strict (≤, ≥)
  2. Graph Boundary Line: Draw line as solid (≤, ≥) or dashed (<, >)
  3. Select Test Point: Choose a point not on the boundary line
  4. Substitute and Evaluate: Plug test point into original inequality
  5. Determine Shading: Shade appropriate half-plane based on test result
  6. Interpret Solution: Understand what the solution region represents
Tip 1: The solution region is always a half-plane (except when the line is vertical or horizontal).
Tip 2: For y > mx + b, the solution is above the line; for y < mx + b, below the line.
Tip 3: Points on solid boundary lines are included in the solution; points on dashed lines are excluded.
Tip 4: Always verify your solution by testing a point in the shaded region.
Applications: Business (production constraints), economics (budget constraints), engineering (resource allocation), science (experimental bounds).
Properties: Solution region is convex, boundary line divides plane into two half-planes, solution set is unbounded.
Essential formulas:

• Boundary line: \(ax + by = c\)

• Line type: Strict → dashed, Non-strict → solid

• Shading: Test point method determines which side to shade

• Half-plane: Region on one side of the boundary line

Linear Inequalities Visualization
Exercise 6: Different Inequality Types
Compare these inequalities:
y < x + 2 (strict inequality)
y ≥ -x + 1 (non-strict inequality)
2x + 3y ≤ 6 (standard form)

Analysis: The chart shows how different inequality types create different solution regions.

  • Strict inequalities (y < x + 2): Dashed boundary line, exclude boundary
  • Non-strict inequalities (y ≥ -x + 1): Solid boundary line, include boundary
  • Standard form (2x + 3y ≤ 6): Convert to slope-intercept to graph

Questions & Answers

Question: How do I know whether to use a solid or dashed line when graphing linear inequalities?

Answer: The type of line depends on the inequality symbol:

  • Solid Line (—): Used for ≤ (less than or equal to) and ≥ (greater than or equal to)
  • Dashed Line (--): Used for < (less than) and > (greater than)

The reason is:

  • Solid line means the points ON the line are included in the solution set
  • Dashed line means the points ON the line are NOT included in the solution set

For example:

  • \(y \leq 2x + 3\): Solid line because points on \(y = 2x + 3\) satisfy the inequality
  • \(y < 2x + 3\): Dashed line because points on \(y = 2x + 3\) do NOT satisfy the inequality

Remember: Equal signs (≤, ≥) → Solid line; No equal signs (<, >) → Dashed line.

Question: How do I choose which side to shade when graphing an inequality?

Answer: Use the test point method:

  1. Choose a point that is NOT on the boundary line (usually the origin (0,0) if it's not on the line)
  2. Substitute the coordinates of the test point into the original inequality
  3. If the inequality is true, shade the side containing the test point
  4. If the inequality is false, shade the opposite side

For example, for the inequality \(y > 2x - 1\):

  • Test point: (0, 0)
  • Substitute: \(0 > 2(0) - 1\) → \(0 > -1\) (true)
  • Since true, shade the side containing (0, 0)

If the test point gives a false result, you shade the other side. This method works for any inequality and ensures accuracy.

Question: What happens when I have a vertical or horizontal line?

Answer: Special cases arise with vertical and horizontal lines:

Vertical Lines: Equations of the form \(x = a\)

  • \(x < a\): Shade to the left of the line
  • \(x \leq a\): Shade to the left, including the line (solid)
  • \(x > a\): Shade to the right of the line
  • \(x \geq a\): Shade to the right, including the line (solid)

Horizontal Lines: Equations of the form \(y = b\)

  • \(y < b\): Shade below the line
  • \(y \leq b\): Shade below, including the line (solid)
  • \(y > b\): Shade above the line
  • \(y \geq b\): Shade above, including the line (solid)

For example, for \(x \geq 3\), draw a solid vertical line at \(x = 3\) and shade to the right. For \(y < 2\), draw a dashed horizontal line at \(y = 2\) and shade below.

Question: How can I check if my graph is correct?

Answer: Use these verification methods:

  1. Test Point Method: Pick a point in the shaded region and verify it satisfies the original inequality
  2. Boundary Check: Verify that points on the boundary line (if included) satisfy the inequality
  3. Opposite Side: Pick a point in the unshaded region and verify it does NOT satisfy the inequality
  4. Visual Inspection: Ensure the line type matches the inequality symbol (solid/dashed)

For example, if you graphed \(y \leq x + 1\):

  • Test a point in shaded region: (0, 0) → \(0 \leq 0 + 1\) → \(0 \leq 1\) ✓
  • Test a point on boundary: (1, 2) → \(2 \leq 1 + 1\) → \(2 \leq 2\) ✓
  • Test a point in unshaded region: (0, 3) → \(3 \leq 0 + 1\) → \(3 \leq 1\) ✗

This systematic verification ensures your graph is accurate.