Two-Variable Linear Inequalities - Ninth Grade Math
Introduction to Two-Variable Linear Inequalities
Linear Inequality in Two Variables: An inequality that can be written in one of these forms:
• $ax + by < c$
• $ax + by > c$
• $ax + by \leq c$
• $ax + by \geq c$
where $a$, $b$, and $c$ are real numbers, and $a$ and $b$ are not both zero
• $ax + by < c$
• $ax + by > c$
• $ax + by \leq c$
• $ax + by \geq c$
where $a$, $b$, and $c$ are real numbers, and $a$ and $b$ are not both zero
Four Inequality Symbols:
• $<$ (less than) - strict inequality
• $>$ (greater than) - strict inequality
• $\leq$ (less than or equal to) - non-strict inequality
• $\geq$ (greater than or equal to) - non-strict inequality
• $<$ (less than) - strict inequality
• $>$ (greater than) - strict inequality
• $\leq$ (less than or equal to) - non-strict inequality
• $\geq$ (greater than or equal to) - non-strict inequality
Key Difference from Equations:
• Equation: Has exactly one solution or infinitely many on a line
• Inequality: Has infinitely many solutions in a region (half-plane)
Example:
• Equation $y = 2x + 1$ → solutions on a line
• Inequality $y > 2x + 1$ → solutions in entire region above the line
• Equation: Has exactly one solution or infinitely many on a line
• Inequality: Has infinitely many solutions in a region (half-plane)
Example:
• Equation $y = 2x + 1$ → solutions on a line
• Inequality $y > 2x + 1$ → solutions in entire region above the line
1. Does (x, y) Satisfy the Inequality?
Steps to Check if $(x, y)$ Satisfies an Inequality:
Step 1: Substitute the x-value into the inequality
Step 2: Substitute the y-value into the inequality
Step 3: Simplify both sides
Step 4: Check if the inequality statement is true
• If TRUE → $(x, y)$ IS a solution ✓
• If FALSE → $(x, y)$ is NOT a solution ✗
Step 1: Substitute the x-value into the inequality
Step 2: Substitute the y-value into the inequality
Step 3: Simplify both sides
Step 4: Check if the inequality statement is true
• If TRUE → $(x, y)$ IS a solution ✓
• If FALSE → $(x, y)$ is NOT a solution ✗
Example 1: Does $(3, 5)$ satisfy $2x + y < 12$?
Substitute:
$2(3) + 5 < 12$
$6 + 5 < 12$
$11 < 12$ ✓ TRUE
Answer: YES, $(3, 5)$ is a solution
Substitute:
$2(3) + 5 < 12$
$6 + 5 < 12$
$11 < 12$ ✓ TRUE
Answer: YES, $(3, 5)$ is a solution
Example 2: Does $(4, 2)$ satisfy $y \geq 3x - 7$?
Substitute:
$2 \geq 3(4) - 7$
$2 \geq 12 - 7$
$2 \geq 5$ ✗ FALSE
Answer: NO, $(4, 2)$ is NOT a solution
Substitute:
$2 \geq 3(4) - 7$
$2 \geq 12 - 7$
$2 \geq 5$ ✗ FALSE
Answer: NO, $(4, 2)$ is NOT a solution
Example 3: Does $(1, 4)$ satisfy $-3x + 2y > 5$?
Substitute:
$-3(1) + 2(4) > 5$
$-3 + 8 > 5$
$5 > 5$ ✗ FALSE (5 is not greater than 5)
Answer: NO
Substitute:
$-3(1) + 2(4) > 5$
$-3 + 8 > 5$
$5 > 5$ ✗ FALSE (5 is not greater than 5)
Answer: NO
Example 4: Does $(2, 6)$ satisfy $y \leq 2x + 2$?
Substitute:
$6 \leq 2(2) + 2$
$6 \leq 6$ ✓ TRUE (6 equals 6)
Answer: YES, $(2, 6)$ is a solution
Substitute:
$6 \leq 2(2) + 2$
$6 \leq 6$ ✓ TRUE (6 equals 6)
Answer: YES, $(2, 6)$ is a solution
Important:
• For $\leq$ or $\geq$: The boundary line IS included (equality allowed)
• For $<$ or $>$: The boundary line is NOT included (strict inequality)
• For $\leq$ or $\geq$: The boundary line IS included (equality allowed)
• For $<$ or $>$: The boundary line is NOT included (strict inequality)
2. Linear Inequalities: Solve for Y
Solving for y: Isolating y on one side to write inequality in slope-intercept form
Purpose: Makes graphing easier and shows relationship clearly
Purpose: Makes graphing easier and shows relationship clearly
Target Form:
• $y < mx + b$
• $y > mx + b$
• $y \leq mx + b$
• $y \geq mx + b$
• $y < mx + b$
• $y > mx + b$
• $y \leq mx + b$
• $y \geq mx + b$
CRITICAL RULE: Multiplying/Dividing by Negatives
When you multiply or divide both sides by a negative number:
REVERSE THE INEQUALITY SIGN!
Examples:
• $-2x < 6$ → divide by -2 → $x > -3$ (sign flipped!)
• $-y \geq 4$ → multiply by -1 → $y \leq -4$ (sign flipped!)
When you multiply or divide both sides by a negative number:
REVERSE THE INEQUALITY SIGN!
Examples:
• $-2x < 6$ → divide by -2 → $x > -3$ (sign flipped!)
• $-y \geq 4$ → multiply by -1 → $y \leq -4$ (sign flipped!)
Steps to Solve for y:
Step 1: Get all y-terms on left side
Step 2: Get all other terms on right side
Step 3: Divide by coefficient of y
Step 4: If dividing by negative → FLIP the inequality sign
Step 5: Simplify
Step 1: Get all y-terms on left side
Step 2: Get all other terms on right side
Step 3: Divide by coefficient of y
Step 4: If dividing by negative → FLIP the inequality sign
Step 5: Simplify
Example 1: Solve for y: $2x + y < 8$
Subtract 2x from both sides:
$y < -2x + 8$
Answer: $y < -2x + 8$
Subtract 2x from both sides:
$y < -2x + 8$
Answer: $y < -2x + 8$
Example 2: Solve for y: $3x - y > 5$
Subtract 3x:
$-y > -3x + 5$
Multiply by -1 (FLIP SIGN):
$y < 3x - 5$
Answer: $y < 3x - 5$
Subtract 3x:
$-y > -3x + 5$
Multiply by -1 (FLIP SIGN):
$y < 3x - 5$
Answer: $y < 3x - 5$
Example 3: Solve for y: $4x + 2y \geq 12$
Subtract 4x:
$2y \geq -4x + 12$
Divide by 2 (positive, no flip):
$y \geq -2x + 6$
Answer: $y \geq -2x + 6$
Subtract 4x:
$2y \geq -4x + 12$
Divide by 2 (positive, no flip):
$y \geq -2x + 6$
Answer: $y \geq -2x + 6$
Example 4: Solve for y: $-6x - 3y < 9$
Add 6x:
$-3y < 6x + 9$
Divide by -3 (FLIP SIGN):
$y > -2x - 3$
Answer: $y > -2x - 3$
Add 6x:
$-3y < 6x + 9$
Divide by -3 (FLIP SIGN):
$y > -2x - 3$
Answer: $y > -2x - 3$
3. Graph a Two-Variable Linear Inequality
Boundary Line: The line formed by replacing the inequality with equality
Half-Plane: The region on one side of the boundary line
Solution Region: The shaded area containing all solutions
Half-Plane: The region on one side of the boundary line
Solution Region: The shaded area containing all solutions
Line Type Rules:
• Solid Line (—): Use for $\leq$ or $\geq$ (boundary included)
• Dashed Line (- - -): Use for $<$ or $>$ (boundary NOT included)
• Solid Line (—): Use for $\leq$ or $\geq$ (boundary included)
• Dashed Line (- - -): Use for $<$ or $>$ (boundary NOT included)
Steps to Graph a Two-Variable Linear Inequality:
Step 1: Solve the inequality for y (if not already)
Step 2: Graph the boundary line
• Replace inequality with = to get equation
• Use solid line for $\leq$ or $\geq$
• Use dashed line for $<$ or $>$
Step 3: Choose a test point (usually $(0, 0)$ if not on line)
Step 4: Substitute test point into original inequality
Step 5: Shade the appropriate region
• If test point satisfies inequality → shade that side
• If test point doesn't satisfy → shade opposite side
Step 1: Solve the inequality for y (if not already)
Step 2: Graph the boundary line
• Replace inequality with = to get equation
• Use solid line for $\leq$ or $\geq$
• Use dashed line for $<$ or $>$
Step 3: Choose a test point (usually $(0, 0)$ if not on line)
Step 4: Substitute test point into original inequality
Step 5: Shade the appropriate region
• If test point satisfies inequality → shade that side
• If test point doesn't satisfy → shade opposite side
Quick Shading Rules (when solved for y):
• $y > mx + b$ → shade ABOVE the line
• $y < mx + b$ → shade BELOW the line
• $y \geq mx + b$ → shade ABOVE (include line)
• $y \leq mx + b$ → shade BELOW (include line)
• $y > mx + b$ → shade ABOVE the line
• $y < mx + b$ → shade BELOW the line
• $y \geq mx + b$ → shade ABOVE (include line)
• $y \leq mx + b$ → shade BELOW (include line)
Example 1: Graph $y \geq 2x - 3$
Step 1: Already solved for y ✓
Step 2: Graph boundary line $y = 2x - 3$
• Y-intercept: $(0, -3)$
• Slope: $m = 2$ (up 2, right 1)
• Draw SOLID line (because $\geq$)
Step 3: Test point: $(0, 0)$
$0 \geq 2(0) - 3$
$0 \geq -3$ ✓ TRUE
Step 4: Shade region including $(0, 0)$ (above the line)
OR use quick rule: $y \geq ...$ means shade ABOVE ✓
Step 1: Already solved for y ✓
Step 2: Graph boundary line $y = 2x - 3$
• Y-intercept: $(0, -3)$
• Slope: $m = 2$ (up 2, right 1)
• Draw SOLID line (because $\geq$)
Step 3: Test point: $(0, 0)$
$0 \geq 2(0) - 3$
$0 \geq -3$ ✓ TRUE
Step 4: Shade region including $(0, 0)$ (above the line)
OR use quick rule: $y \geq ...$ means shade ABOVE ✓
Example 2: Graph $y < -x + 4$
Boundary line: $y = -x + 4$
• Y-intercept: $(0, 4)$
• Slope: $m = -1$
• Draw DASHED line (because $<$)
Quick rule: $y < ...$ means shade BELOW the line
Verify with test point $(0, 0)$:
$0 < -(0) + 4$ → $0 < 4$ ✓ TRUE
Shade side with $(0, 0)$ ✓
Boundary line: $y = -x + 4$
• Y-intercept: $(0, 4)$
• Slope: $m = -1$
• Draw DASHED line (because $<$)
Quick rule: $y < ...$ means shade BELOW the line
Verify with test point $(0, 0)$:
$0 < -(0) + 4$ → $0 < 4$ ✓ TRUE
Shade side with $(0, 0)$ ✓
Example 3: Graph $2x + y > 4$
Step 1: Solve for y
$y > -2x + 4$
Step 2: Boundary line: $y = -2x + 4$
• DASHED line (because $>$)
• Y-intercept: $(0, 4)$
• Slope: $-2$
Step 3: Shade ABOVE the line ($y > ...$)
Step 1: Solve for y
$y > -2x + 4$
Step 2: Boundary line: $y = -2x + 4$
• DASHED line (because $>$)
• Y-intercept: $(0, 4)$
• Slope: $-2$
Step 3: Shade ABOVE the line ($y > ...$)
Common Mistakes to Avoid:
• Forgetting to flip inequality when dividing by negative
• Using wrong line type (solid vs. dashed)
• Shading wrong side
• Not checking if test point is on the boundary line
• Forgetting to flip inequality when dividing by negative
• Using wrong line type (solid vs. dashed)
• Shading wrong side
• Not checking if test point is on the boundary line
4. Write a Linear Inequality from a Graph
Steps to Write Inequality from Graph:
Step 1: Identify the boundary line
• Find y-intercept $(0, b)$
• Find slope $m$ using two points
Step 2: Write equation of boundary line: $y = mx + b$
Step 3: Determine line type
• Solid line → use $\leq$ or $\geq$
• Dashed line → use $<$ or $>$
Step 4: Determine which inequality sign
• Shaded above → use $\geq$ or $>$
• Shaded below → use $\leq$ or $<$
Step 5: Write final inequality
Step 6: Verify with test point from shaded region
Step 1: Identify the boundary line
• Find y-intercept $(0, b)$
• Find slope $m$ using two points
Step 2: Write equation of boundary line: $y = mx + b$
Step 3: Determine line type
• Solid line → use $\leq$ or $\geq$
• Dashed line → use $<$ or $>$
Step 4: Determine which inequality sign
• Shaded above → use $\geq$ or $>$
• Shaded below → use $\leq$ or $<$
Step 5: Write final inequality
Step 6: Verify with test point from shaded region
Example 1: Graph shows:
• Dashed line through $(0, 2)$ and $(2, 6)$
• Shaded region above the line
Find slope:
$m = \frac{6-2}{2-0} = \frac{4}{2} = 2$
Y-intercept: $b = 2$
Boundary equation: $y = 2x + 2$
Determine inequality:
• Dashed line → strict inequality ($<$ or $>$)
• Shaded above → use $>$
Answer: $y > 2x + 2$
• Dashed line through $(0, 2)$ and $(2, 6)$
• Shaded region above the line
Find slope:
$m = \frac{6-2}{2-0} = \frac{4}{2} = 2$
Y-intercept: $b = 2$
Boundary equation: $y = 2x + 2$
Determine inequality:
• Dashed line → strict inequality ($<$ or $>$)
• Shaded above → use $>$
Answer: $y > 2x + 2$
Example 2: Graph shows:
• Solid line through $(0, -1)$ and $(3, 5)$
• Shaded region below the line
Slope: $m = \frac{5-(-1)}{3-0} = \frac{6}{3} = 2$
Equation: $y = 2x - 1$
Determine inequality:
• Solid line → use $\leq$ or $\geq$
• Shaded below → use $\leq$
Answer: $y \leq 2x - 1$
• Solid line through $(0, -1)$ and $(3, 5)$
• Shaded region below the line
Slope: $m = \frac{5-(-1)}{3-0} = \frac{6}{3} = 2$
Equation: $y = 2x - 1$
Determine inequality:
• Solid line → use $\leq$ or $\geq$
• Shaded below → use $\leq$
Answer: $y \leq 2x - 1$
Example 3: Horizontal line at $y = 3$, solid, shaded above
Answer: $y \geq 3$
Answer: $y \geq 3$
Example 4: Vertical line at $x = -2$, dashed, shaded to the right
Answer: $x > -2$
Answer: $x > -2$
5. Write Two-Variable Inequalities: Word Problems
Key Words and Phrases:
• "at least" → $\geq$
• "at most" → $\leq$
• "more than" → $>$
• "less than" → $<$
• "no more than" → $\leq$
• "no less than" → $\geq$
• "minimum" → $\geq$
• "maximum" → $\leq$
• "exceeds" → $>$
• "cannot exceed" → $\leq$
• "at least" → $\geq$
• "at most" → $\leq$
• "more than" → $>$
• "less than" → $<$
• "no more than" → $\leq$
• "no less than" → $\geq$
• "minimum" → $\geq$
• "maximum" → $\leq$
• "exceeds" → $>$
• "cannot exceed" → $\leq$
Steps for Word Problems:
Step 1: Define variables clearly
Step 2: Identify constraints and relationships
Step 3: Translate words into mathematical expressions
Step 4: Determine the correct inequality symbol
Step 5: Write the inequality
Step 6: Check if it makes sense in context
Step 1: Define variables clearly
Step 2: Identify constraints and relationships
Step 3: Translate words into mathematical expressions
Step 4: Determine the correct inequality symbol
Step 5: Write the inequality
Step 6: Check if it makes sense in context
Example 1: Adult tickets cost $10 and child tickets cost $6. A family wants to spend at most $50. Write an inequality.
Define variables:
• Let $a$ = number of adult tickets
• Let $c$ = number of child tickets
Cost equation: $10a + 6c$
"at most $50" means $\leq 50$
Answer: $10a + 6c \leq 50$
Define variables:
• Let $a$ = number of adult tickets
• Let $c$ = number of child tickets
Cost equation: $10a + 6c$
"at most $50" means $\leq 50$
Answer: $10a + 6c \leq 50$
Example 2: A store sells apples for $2/lb and oranges for $3/lb. You want to buy fruit but spend more than $15. Write inequality.
Variables:
• $x$ = pounds of apples
• $y$ = pounds of oranges
"more than $15" → $>$
Answer: $2x + 3y > 15$
Variables:
• $x$ = pounds of apples
• $y$ = pounds of oranges
"more than $15" → $>$
Answer: $2x + 3y > 15$
Example 3: A truck can carry at most 5000 pounds. Small boxes weigh 50 lbs and large boxes weigh 80 lbs. Write inequality.
Variables:
• $s$ = number of small boxes
• $l$ = number of large boxes
"at most 5000" → $\leq 5000$
Answer: $50s + 80l \leq 5000$
Variables:
• $s$ = number of small boxes
• $l$ = number of large boxes
"at most 5000" → $\leq 5000$
Answer: $50s + 80l \leq 5000$
Example 4: To maintain health, you should do at least 30 minutes of cardio and strength training combined daily. Write inequality.
Variables:
• $c$ = minutes of cardio
• $s$ = minutes of strength training
"at least 30" → $\geq 30$
Answer: $c + s \geq 30$
Variables:
• $c$ = minutes of cardio
• $s$ = minutes of strength training
"at least 30" → $\geq 30$
Answer: $c + s \geq 30$
6-7. Systems of Linear Inequalities
System of Linear Inequalities: Two or more linear inequalities with the same variables
Solution: An ordered pair $(x, y)$ that satisfies ALL inequalities in the system
Solution Region: The area where all shaded regions overlap
Solution: An ordered pair $(x, y)$ that satisfies ALL inequalities in the system
Solution Region: The area where all shaded regions overlap
6. Is (x, y) a Solution to the System?
Steps to Check if $(x, y)$ is a Solution to System:
Step 1: Substitute $(x, y)$ into FIRST inequality
Step 2: Check if it's true
Step 3: Substitute $(x, y)$ into SECOND inequality
Step 4: Check if it's true
Step 5: Repeat for all inequalities
Result: Point is solution ONLY if ALL inequalities are satisfied
Step 1: Substitute $(x, y)$ into FIRST inequality
Step 2: Check if it's true
Step 3: Substitute $(x, y)$ into SECOND inequality
Step 4: Check if it's true
Step 5: Repeat for all inequalities
Result: Point is solution ONLY if ALL inequalities are satisfied
Example 1: Is $(3, 2)$ a solution?
$$\begin{cases} y < 2x - 1 \\ y \geq -x + 4 \end{cases}$$
Check First Inequality:
$2 < 2(3) - 1$
$2 < 6 - 1$
$2 < 5$ ✓ TRUE
Check Second Inequality:
$2 \geq -(3) + 4$
$2 \geq 1$ ✓ TRUE
Answer: YES, $(3, 2)$ is a solution (both are satisfied)
$$\begin{cases} y < 2x - 1 \\ y \geq -x + 4 \end{cases}$$
Check First Inequality:
$2 < 2(3) - 1$
$2 < 6 - 1$
$2 < 5$ ✓ TRUE
Check Second Inequality:
$2 \geq -(3) + 4$
$2 \geq 1$ ✓ TRUE
Answer: YES, $(3, 2)$ is a solution (both are satisfied)
Example 2: Is $(1, 5)$ a solution?
$$\begin{cases} x + y \leq 8 \\ 2x - y > 0 \end{cases}$$
First: $1 + 5 \leq 8$ → $6 \leq 8$ ✓
Second: $2(1) - 5 > 0$ → $2 - 5 > 0$ → $-3 > 0$ ✗
Answer: NO (second inequality not satisfied)
$$\begin{cases} x + y \leq 8 \\ 2x - y > 0 \end{cases}$$
First: $1 + 5 \leq 8$ → $6 \leq 8$ ✓
Second: $2(1) - 5 > 0$ → $2 - 5 > 0$ → $-3 > 0$ ✗
Answer: NO (second inequality not satisfied)
7. Solve Systems by Graphing
Steps to Solve System of Inequalities by Graphing:
Step 1: Graph the first inequality
• Draw boundary line (solid or dashed)
• Shade appropriate region
Step 2: Graph the second inequality on same axes
• Draw boundary line
• Shade appropriate region
Step 3: Identify the overlap region
• Solution is where BOTH shadings overlap
• Usually shown with darker/double shading
Step 4: Verify with test point from overlap region
Step 1: Graph the first inequality
• Draw boundary line (solid or dashed)
• Shade appropriate region
Step 2: Graph the second inequality on same axes
• Draw boundary line
• Shade appropriate region
Step 3: Identify the overlap region
• Solution is where BOTH shadings overlap
• Usually shown with darker/double shading
Step 4: Verify with test point from overlap region
Types of Solution Regions:
• Overlapping region exists: System has infinitely many solutions
• No overlap: System has no solution
• Entire plane: Every point is a solution (rare)
• Overlapping region exists: System has infinitely many solutions
• No overlap: System has no solution
• Entire plane: Every point is a solution (rare)
Example 1: Solve by graphing
$$\begin{cases} y \geq x + 1 \\ y < -x + 5 \end{cases}$$
Graph First Inequality: $y \geq x + 1$
• Boundary: $y = x + 1$ (solid line)
• Y-intercept: $(0, 1)$, slope: 1
• Shade above
Graph Second Inequality: $y < -x + 5$
• Boundary: $y = -x + 5$ (dashed line)
• Y-intercept: $(0, 5)$, slope: -1
• Shade below
Solution: Overlapping region (darker shaded area)
• This is a bounded region (triangle-like shape)
• Any point in this region is a solution
$$\begin{cases} y \geq x + 1 \\ y < -x + 5 \end{cases}$$
Graph First Inequality: $y \geq x + 1$
• Boundary: $y = x + 1$ (solid line)
• Y-intercept: $(0, 1)$, slope: 1
• Shade above
Graph Second Inequality: $y < -x + 5$
• Boundary: $y = -x + 5$ (dashed line)
• Y-intercept: $(0, 5)$, slope: -1
• Shade below
Solution: Overlapping region (darker shaded area)
• This is a bounded region (triangle-like shape)
• Any point in this region is a solution
Example 2: Solve the system
$$\begin{cases} 2x + y \leq 6 \\ x - y > 2 \end{cases}$$
Solve first for y:
• $y \leq -2x + 6$ (solid line, shade below)
• $y < x - 2$ (dashed line, shade below)
Solution: Region where both shadings overlap
$$\begin{cases} 2x + y \leq 6 \\ x - y > 2 \end{cases}$$
Solve first for y:
• $y \leq -2x + 6$ (solid line, shade below)
• $y < x - 2$ (dashed line, shade below)
Solution: Region where both shadings overlap
Example 3 (Word Problem): A bakery makes cookies and cakes. Cookies take 2 hours, cakes take 3 hours. They have at most 12 hours. They want to make at least 2 cookies. Graph constraints.
Variables: $c$ = cookies, $k$ = cakes
System:
$$\begin{cases} 2c + 3k \leq 12 \text{ (time constraint)} \\ c \geq 2 \text{ (minimum cookies)} \\ c \geq 0, k \geq 0 \text{ (non-negative)} \end{cases}$$
Graph all inequalities and find overlap region
Variables: $c$ = cookies, $k$ = cakes
System:
$$\begin{cases} 2c + 3k \leq 12 \text{ (time constraint)} \\ c \geq 2 \text{ (minimum cookies)} \\ c \geq 0, k \geq 0 \text{ (non-negative)} \end{cases}$$
Graph all inequalities and find overlap region
Graphing Tips for Systems:
• Use different colors for each inequality (if possible)
• Label each boundary line
• Use distinct patterns for different shadings
• The solution region is where ALL conditions are met
• Corner points (vertices) are often important in applications
• Use different colors for each inequality (if possible)
• Label each boundary line
• Use distinct patterns for different shadings
• The solution region is where ALL conditions are met
• Corner points (vertices) are often important in applications
Special Cases and Important Notes
Horizontal and Vertical Inequalities:
Horizontal Lines:
• $y > k$ → shade ABOVE horizontal line at $y = k$
• $y < k$ → shade BELOW horizontal line at $y = k$
• $y \geq k$ → solid line, shade above
• $y \leq k$ → solid line, shade below
Vertical Lines:
• $x > h$ → shade RIGHT of vertical line at $x = h$
• $x < h$ → shade LEFT of vertical line at $x = h$
• $x \geq h$ → solid line, shade right
• $x \leq h$ → solid line, shade left
Horizontal Lines:
• $y > k$ → shade ABOVE horizontal line at $y = k$
• $y < k$ → shade BELOW horizontal line at $y = k$
• $y \geq k$ → solid line, shade above
• $y \leq k$ → solid line, shade below
Vertical Lines:
• $x > h$ → shade RIGHT of vertical line at $x = h$
• $x < h$ → shade LEFT of vertical line at $x = h$
• $x \geq h$ → solid line, shade right
• $x \leq h$ → solid line, shade left
Example: Graph $y > 3$
• Draw horizontal dashed line at $y = 3$
• Shade entire region above the line
• All points with y-coordinate greater than 3 are solutions
• Draw horizontal dashed line at $y = 3$
• Shade entire region above the line
• All points with y-coordinate greater than 3 are solutions
Example: Graph system
$$\begin{cases} x \geq -2 \\ y < 4 \end{cases}$$
• Vertical solid line at $x = -2$, shade right
• Horizontal dashed line at $y = 4$, shade below
• Solution: Rectangular region to right of $x = -2$ and below $y = 4$
$$\begin{cases} x \geq -2 \\ y < 4 \end{cases}$$
• Vertical solid line at $x = -2$, shade right
• Horizontal dashed line at $y = 4$, shade below
• Solution: Rectangular region to right of $x = -2$ and below $y = 4$
Quick Reference Guide
| Symbol | Meaning | Line Type | Shading (when $y$ form) |
|---|---|---|---|
| $<$ | Less than | Dashed | Below |
| $>$ | Greater than | Dashed | Above |
| $\leq$ | Less than or equal | Solid | Below (include line) |
| $\geq$ | Greater than or equal | Solid | Above (include line) |
Key Differences: Equations vs. Inequalities
| Aspect | Linear Equation | Linear Inequality |
|---|---|---|
| Symbol | $=$ | $<, >, \leq, \geq$ |
| Graph | Single line | Line + shaded region |
| Solutions | Points on line | Points in region |
| Number of Solutions | Infinitely many (on line) | Infinitely many (in region) |
Word Problem Key Phrases:
• "at least" = $\geq$
• "at most" = $\leq$
• "more than" = $>$
• "less than" = $<$
• "minimum" = $\geq$
• "maximum" = $\leq$
• "no more than" = $\leq$
• "no less than" = $\geq$
• "at least" = $\geq$
• "at most" = $\leq$
• "more than" = $>$
• "less than" = $<$
• "minimum" = $\geq$
• "maximum" = $\leq$
• "no more than" = $\leq$
• "no less than" = $\geq$
Critical Rules to Remember:
1. Multiplying/dividing by negative → FLIP inequality sign
2. Solid line for $\leq$ or $\geq$ (boundary included)
3. Dashed line for $<$ or $>$ (boundary excluded)
4. When $y > mx + b$: shade ABOVE
5. When $y < mx + b$: shade BELOW
6. System solution: ALL inequalities must be satisfied
7. Test points verify correctness
1. Multiplying/dividing by negative → FLIP inequality sign
2. Solid line for $\leq$ or $\geq$ (boundary included)
3. Dashed line for $<$ or $>$ (boundary excluded)
4. When $y > mx + b$: shade ABOVE
5. When $y < mx + b$: shade BELOW
6. System solution: ALL inequalities must be satisfied
7. Test points verify correctness
Common Applications
Real-World Uses of Linear Inequalities:
1. Budget Constraints:
"Spending cannot exceed a certain amount"
Example: $3x + 5y \leq 100$
2. Nutrition Requirements:
"Must consume at least certain nutrients"
Example: $2x + 4y \geq 50$
3. Time Management:
"Total time available is limited"
Example: $x + y \leq 24$
4. Production Limits:
"Factory capacity constraints"
Example: $5x + 3y \leq 1000$
5. Safety Requirements:
"Weight limits, minimum standards"
Example: $w \geq 150$, $h \leq 500$
1. Budget Constraints:
"Spending cannot exceed a certain amount"
Example: $3x + 5y \leq 100$
2. Nutrition Requirements:
"Must consume at least certain nutrients"
Example: $2x + 4y \geq 50$
3. Time Management:
"Total time available is limited"
Example: $x + y \leq 24$
4. Production Limits:
"Factory capacity constraints"
Example: $5x + 3y \leq 1000$
5. Safety Requirements:
"Weight limits, minimum standards"
Example: $w \geq 150$, $h \leq 500$
Success Tips for Linear Inequalities:
✓ Always check if you need to flip the inequality sign
✓ Use solid line for $\leq$ or $\geq$, dashed for $<$ or $>$
✓ Test a point to verify which side to shade
✓ For systems, find where ALL shadings overlap
✓ Label graphs clearly with inequality symbols
✓ Read word problems carefully for key phrases
✓ Verify solutions by substituting back
✓ Remember: solution region has infinitely many points
✓ Check boundary points separately for inclusion
✓ Practice translating words into mathematical inequalities
✓ Always check if you need to flip the inequality sign
✓ Use solid line for $\leq$ or $\geq$, dashed for $<$ or $>$
✓ Test a point to verify which side to shade
✓ For systems, find where ALL shadings overlap
✓ Label graphs clearly with inequality symbols
✓ Read word problems carefully for key phrases
✓ Verify solutions by substituting back
✓ Remember: solution region has infinitely many points
✓ Check boundary points separately for inclusion
✓ Practice translating words into mathematical inequalities