Solve Linear Inequalities - Ninth Grade Math
Introduction to Linear Inequalities
Inequality: A mathematical statement that compares two expressions using inequality symbols
Linear Inequality: An inequality involving variables with exponent 1
Solution: Any value that makes the inequality true
Solution Set: All values that satisfy the inequality
Linear Inequality: An inequality involving variables with exponent 1
Solution: Any value that makes the inequality true
Solution Set: All values that satisfy the inequality
Inequality Symbols
| Symbol | Meaning | Alternative Phrases | Example |
|---|---|---|---|
| < | Less than | Fewer than, below | $x < 5$ |
| > | Greater than | More than, above | $x > 3$ |
| ≤ | Less than or equal to | At most, no more than, maximum | $x \leq 7$ |
| ≥ | Greater than or equal to | At least, no less than, minimum | $x \geq 2$ |
| ≠ | Not equal to | Does not equal | $x \neq 0$ |
1. Graph Inequalities
Graphing on a Number Line: Visual representation of solution sets
Open Circle (○): Used for < and > (number NOT included)
Closed/Filled Circle (●): Used for ≤ and ≥ (number IS included)
Shading/Arrow: Shows all values in the solution set
Open Circle (○): Used for < and > (number NOT included)
Closed/Filled Circle (●): Used for ≤ and ≥ (number IS included)
Shading/Arrow: Shows all values in the solution set
Graphing Rules:
• For $x < a$ or $x \leq a$: Shade to the LEFT (smaller values)
• For $x > a$ or $x \geq a$: Shade to the RIGHT (larger values)
• Open circle: Number is NOT included ($<$ or $>$)
• Closed circle: Number IS included ($\leq$ or $\geq$)
• For $x < a$ or $x \leq a$: Shade to the LEFT (smaller values)
• For $x > a$ or $x \geq a$: Shade to the RIGHT (larger values)
• Open circle: Number is NOT included ($<$ or $>$)
• Closed circle: Number IS included ($\leq$ or $\geq$)
| Inequality | Circle Type | Shading Direction | Description |
|---|---|---|---|
| $x < 3$ | Open circle at 3 | ← Left | All numbers less than 3 |
| $x > 3$ | Open circle at 3 | Right → | All numbers greater than 3 |
| $x \leq 3$ | Closed circle at 3 | ← Left | All numbers less than or equal to 3 |
| $x \geq 3$ | Closed circle at 3 | Right → | All numbers greater than or equal to 3 |
Example 1: Graph $x > 2$ on a number line.
Solution:
• Draw a number line
• Place an OPEN circle at 2
• Shade to the RIGHT (toward larger numbers)
• This represents all numbers greater than 2
Solution:
• Draw a number line
• Place an OPEN circle at 2
• Shade to the RIGHT (toward larger numbers)
• This represents all numbers greater than 2
Example 2: Graph $x \leq -1$ on a number line.
Solution:
• Draw a number line
• Place a CLOSED circle at -1
• Shade to the LEFT (toward smaller numbers)
• This includes -1 and all numbers less than -1
Solution:
• Draw a number line
• Place a CLOSED circle at -1
• Shade to the LEFT (toward smaller numbers)
• This includes -1 and all numbers less than -1
2. Write Inequalities from Graphs
Steps to Write Inequalities from Graphs:
Step 1: Identify the critical point (where circle is placed)
Step 2: Determine if circle is open or closed
• Open → use $<$ or $>$
• Closed → use $\leq$ or $\geq$
Step 3: Determine shading direction
• Left → use $<$ or $\leq$
• Right → use $>$ or $\geq$
Step 4: Write the inequality
Step 1: Identify the critical point (where circle is placed)
Step 2: Determine if circle is open or closed
• Open → use $<$ or $>$
• Closed → use $\leq$ or $\geq$
Step 3: Determine shading direction
• Left → use $<$ or $\leq$
• Right → use $>$ or $\geq$
Step 4: Write the inequality
Example 1: A graph shows a closed circle at 5 with shading to the right.
Solution:
• Closed circle → include the equal sign
• Shading right → greater than
• Answer: $x \geq 5$
Solution:
• Closed circle → include the equal sign
• Shading right → greater than
• Answer: $x \geq 5$
Example 2: A graph shows an open circle at -3 with shading to the left.
Solution:
• Open circle → strict inequality (no equal)
• Shading left → less than
• Answer: $x < -3$
Solution:
• Open circle → strict inequality (no equal)
• Shading left → less than
• Answer: $x < -3$
3. Identify Solutions to Inequalities
Solution: A value that makes the inequality true when substituted
Testing Solutions: Substitute the value and check if the resulting statement is true
Testing Solutions: Substitute the value and check if the resulting statement is true
How to Check if a Value is a Solution:
Step 1: Substitute the value for the variable
Step 2: Simplify both sides
Step 3: Determine if the inequality statement is true
Step 4: If true → it's a solution; if false → not a solution
Step 1: Substitute the value for the variable
Step 2: Simplify both sides
Step 3: Determine if the inequality statement is true
Step 4: If true → it's a solution; if false → not a solution
Example 1: Is $x = 7$ a solution to $2x + 3 < 20$?
Substitute: $2(7) + 3 < 20$
Simplify: $14 + 3 < 20$
Result: $17 < 20$ ✓ TRUE
Answer: Yes, $x = 7$ is a solution.
Substitute: $2(7) + 3 < 20$
Simplify: $14 + 3 < 20$
Result: $17 < 20$ ✓ TRUE
Answer: Yes, $x = 7$ is a solution.
Example 2: Is $x = 5$ a solution to $3x - 4 > 12$?
Substitute: $3(5) - 4 > 12$
Simplify: $15 - 4 > 12$
Result: $11 > 12$ ✗ FALSE
Answer: No, $x = 5$ is NOT a solution.
Substitute: $3(5) - 4 > 12$
Simplify: $15 - 4 > 12$
Result: $11 > 12$ ✗ FALSE
Answer: No, $x = 5$ is NOT a solution.
Example 3: Is $x = -2$ a solution to $5 - 2x \geq 9$?
Substitute: $5 - 2(-2) \geq 9$
Simplify: $5 + 4 \geq 9$
Result: $9 \geq 9$ ✓ TRUE
Answer: Yes, $x = -2$ is a solution.
Substitute: $5 - 2(-2) \geq 9$
Simplify: $5 + 4 \geq 9$
Result: $9 \geq 9$ ✓ TRUE
Answer: Yes, $x = -2$ is a solution.
Rules for Solving Linear Inequalities
Key Rules (Just Like Equations):
1. Addition Property: If $a < b$, then $a + c < b + c$
2. Subtraction Property: If $a < b$, then $a - c < b - c$
3. Multiplication by Positive: If $a < b$ and $c > 0$, then $ac < bc$
4. Division by Positive: If $a < b$ and $c > 0$, then $\frac{a}{c} < \frac{b}{c}$
1. Addition Property: If $a < b$, then $a + c < b + c$
2. Subtraction Property: If $a < b$, then $a - c < b - c$
3. Multiplication by Positive: If $a < b$ and $c > 0$, then $ac < bc$
4. Division by Positive: If $a < b$ and $c > 0$, then $\frac{a}{c} < \frac{b}{c}$
⚠️ CRITICAL RULE - MUST REVERSE INEQUALITY SIGN:
When multiplying or dividing BOTH sides by a NEGATIVE number:
• FLIP the inequality sign
• $<$ becomes $>$
• $>$ becomes $<$
• $\leq$ becomes $\geq$
• $\geq$ becomes $\leq$
Example:
If $-2x < 6$, divide both sides by $-2$:
$x > -3$ (sign REVERSED from $<$ to $>$)
When multiplying or dividing BOTH sides by a NEGATIVE number:
• FLIP the inequality sign
• $<$ becomes $>$
• $>$ becomes $<$
• $\leq$ becomes $\geq$
• $\geq$ becomes $\leq$
Example:
If $-2x < 6$, divide both sides by $-2$:
$x > -3$ (sign REVERSED from $<$ to $>$)
4. Solve One-Step Linear Inequalities: Addition and Subtraction
Addition Inequalities: $x - a < b$
Solution: Add $a$ to both sides → $x < b + a$
Subtraction Inequalities: $x + a < b$
Solution: Subtract $a$ from both sides → $x < b - a$
Solution: Add $a$ to both sides → $x < b + a$
Subtraction Inequalities: $x + a < b$
Solution: Subtract $a$ from both sides → $x < b - a$
Example 1: Solve $x + 7 < 12$
Subtract 7 from both sides:
$x + 7 - 7 < 12 - 7$
$x < 5$
Solution: $x < 5$
(All numbers less than 5)
Subtract 7 from both sides:
$x + 7 - 7 < 12 - 7$
$x < 5$
Solution: $x < 5$
(All numbers less than 5)
Example 2: Solve $x - 5 \geq 3$
Add 5 to both sides:
$x - 5 + 5 \geq 3 + 5$
$x \geq 8$
Solution: $x \geq 8$
(All numbers greater than or equal to 8)
Add 5 to both sides:
$x - 5 + 5 \geq 3 + 5$
$x \geq 8$
Solution: $x \geq 8$
(All numbers greater than or equal to 8)
Example 3: Solve $-4 + x > 10$
Add 4 to both sides:
$x - 4 + 4 > 10 + 4$
$x > 14$
Solution: $x > 14$
Add 4 to both sides:
$x - 4 + 4 > 10 + 4$
$x > 14$
Solution: $x > 14$
5. Solve One-Step Linear Inequalities: Multiplication and Division
Multiplication Inequalities: $\frac{x}{a} < b$
• If $a > 0$: Multiply both sides by $a$ → $x < ab$ (no change)
• If $a < 0$: Multiply both sides by $a$ → $x > ab$ (REVERSE sign)
Division Inequalities: $ax < b$
• If $a > 0$: Divide both sides by $a$ → $x < \frac{b}{a}$ (no change)
• If $a < 0$: Divide both sides by $a$ → $x > \frac{b}{a}$ (REVERSE sign)
• If $a > 0$: Multiply both sides by $a$ → $x < ab$ (no change)
• If $a < 0$: Multiply both sides by $a$ → $x > ab$ (REVERSE sign)
Division Inequalities: $ax < b$
• If $a > 0$: Divide both sides by $a$ → $x < \frac{b}{a}$ (no change)
• If $a < 0$: Divide both sides by $a$ → $x > \frac{b}{a}$ (REVERSE sign)
Example 1: Solve $5x > 20$
Divide both sides by 5 (positive):
$\frac{5x}{5} > \frac{20}{5}$
$x > 4$ (sign stays the same)
Solution: $x > 4$
Divide both sides by 5 (positive):
$\frac{5x}{5} > \frac{20}{5}$
$x > 4$ (sign stays the same)
Solution: $x > 4$
Example 2: Solve $-3x < 15$
Divide both sides by -3 (negative):
$\frac{-3x}{-3} > \frac{15}{-3}$ ← SIGN REVERSED!
$x > -5$
Solution: $x > -5$
Divide both sides by -3 (negative):
$\frac{-3x}{-3} > \frac{15}{-3}$ ← SIGN REVERSED!
$x > -5$
Solution: $x > -5$
Example 3: Solve $\frac{x}{4} \leq 6$
Multiply both sides by 4 (positive):
$4 \cdot \frac{x}{4} \leq 4 \cdot 6$
$x \leq 24$ (sign stays the same)
Solution: $x \leq 24$
Multiply both sides by 4 (positive):
$4 \cdot \frac{x}{4} \leq 4 \cdot 6$
$x \leq 24$ (sign stays the same)
Solution: $x \leq 24$
Example 4: Solve $-\frac{x}{2} \geq 7$
Multiply both sides by -2 (negative):
$-2 \cdot \left(-\frac{x}{2}\right) \leq -2 \cdot 7$ ← SIGN REVERSED!
$x \leq -14$
Solution: $x \leq -14$
Multiply both sides by -2 (negative):
$-2 \cdot \left(-\frac{x}{2}\right) \leq -2 \cdot 7$ ← SIGN REVERSED!
$x \leq -14$
Solution: $x \leq -14$
6. Solve One-Step Linear Inequalities (General)
General Strategy:
1. Identify the operation connecting the variable
2. Use the inverse operation
3. Remember: REVERSE the sign when multiplying/dividing by negatives
4. Write final answer in simplest form
1. Identify the operation connecting the variable
2. Use the inverse operation
3. Remember: REVERSE the sign when multiplying/dividing by negatives
4. Write final answer in simplest form
| Type | Example | Operation | Solution |
|---|---|---|---|
| Addition | $x + 3 < 8$ | Subtract 3 | $x < 5$ |
| Subtraction | $x - 4 > 2$ | Add 4 | $x > 6$ |
| Multiplication (positive) | $3x \leq 12$ | Divide by 3 | $x \leq 4$ |
| Multiplication (negative) | $-2x < 10$ | Divide by -2, FLIP sign | $x > -5$ |
| Division | $\frac{x}{5} \geq 3$ | Multiply by 5 | $x \geq 15$ |
7. Graph Solutions to One-Step Linear Inequalities
Steps to Solve and Graph:
Step 1: Solve the inequality
Step 2: Draw a number line
Step 3: Mark the critical number with appropriate circle
Step 4: Shade in the correct direction
Step 5: Draw an arrow to show continuation
Step 1: Solve the inequality
Step 2: Draw a number line
Step 3: Mark the critical number with appropriate circle
Step 4: Shade in the correct direction
Step 5: Draw an arrow to show continuation
Example 1: Solve and graph $x + 3 \leq 7$
Solve:
$x + 3 - 3 \leq 7 - 3$
$x \leq 4$
Graph:
• Closed circle at 4 (includes 4)
• Shade left (all numbers ≤ 4)
• Draw arrow pointing left
Solve:
$x + 3 - 3 \leq 7 - 3$
$x \leq 4$
Graph:
• Closed circle at 4 (includes 4)
• Shade left (all numbers ≤ 4)
• Draw arrow pointing left
Example 2: Solve and graph $-2x > 6$
Solve:
$\frac{-2x}{-2} < \frac{6}{-2}$ (REVERSE sign!)
$x < -3$
Graph:
• Open circle at -3 (does not include -3)
• Shade left (all numbers < -3)
• Draw arrow pointing left
Solve:
$\frac{-2x}{-2} < \frac{6}{-2}$ (REVERSE sign!)
$x < -3$
Graph:
• Open circle at -3 (does not include -3)
• Shade left (all numbers < -3)
• Draw arrow pointing left
8. Solve Two-Step Linear Inequalities
Two-Step Inequality: Requires two operations to isolate the variable
General Form: $ax + b < c$ or similar variations
General Form: $ax + b < c$ or similar variations
Steps to Solve Two-Step Inequalities:
Step 1: Add or subtract to isolate the term with the variable
Step 2: Multiply or divide to isolate the variable
Step 3: Remember to REVERSE sign if multiplying/dividing by negative
Step 4: Write final answer
Step 1: Add or subtract to isolate the term with the variable
Step 2: Multiply or divide to isolate the variable
Step 3: Remember to REVERSE sign if multiplying/dividing by negative
Step 4: Write final answer
Example 1: Solve $3x + 5 < 14$
Step 1: Subtract 5
$3x + 5 - 5 < 14 - 5$
$3x < 9$
Step 2: Divide by 3
$\frac{3x}{3} < \frac{9}{3}$
$x < 3$
Solution: $x < 3$
Step 1: Subtract 5
$3x + 5 - 5 < 14 - 5$
$3x < 9$
Step 2: Divide by 3
$\frac{3x}{3} < \frac{9}{3}$
$x < 3$
Solution: $x < 3$
Example 2: Solve $-2x + 7 \geq 15$
Step 1: Subtract 7
$-2x + 7 - 7 \geq 15 - 7$
$-2x \geq 8$
Step 2: Divide by -2 (REVERSE sign!)
$\frac{-2x}{-2} \leq \frac{8}{-2}$
$x \leq -4$
Solution: $x \leq -4$
Step 1: Subtract 7
$-2x + 7 - 7 \geq 15 - 7$
$-2x \geq 8$
Step 2: Divide by -2 (REVERSE sign!)
$\frac{-2x}{-2} \leq \frac{8}{-2}$
$x \leq -4$
Solution: $x \leq -4$
Example 3: Solve $\frac{x}{4} - 3 > 2$
Step 1: Add 3
$\frac{x}{4} - 3 + 3 > 2 + 3$
$\frac{x}{4} > 5$
Step 2: Multiply by 4
$4 \cdot \frac{x}{4} > 4 \cdot 5$
$x > 20$
Solution: $x > 20$
Step 1: Add 3
$\frac{x}{4} - 3 + 3 > 2 + 3$
$\frac{x}{4} > 5$
Step 2: Multiply by 4
$4 \cdot \frac{x}{4} > 4 \cdot 5$
$x > 20$
Solution: $x > 20$
9. Graph Solutions to Two-Step Linear Inequalities
Example 1: Solve and graph $5x - 8 \leq 12$
Solve:
Step 1: $5x - 8 + 8 \leq 12 + 8$ → $5x \leq 20$
Step 2: $x \leq 4$
Graph:
• Closed circle at 4
• Shade left
• Solution: all numbers less than or equal to 4
Solve:
Step 1: $5x - 8 + 8 \leq 12 + 8$ → $5x \leq 20$
Step 2: $x \leq 4$
Graph:
• Closed circle at 4
• Shade left
• Solution: all numbers less than or equal to 4
Example 2: Solve and graph $-3x + 2 < 11$
Solve:
Step 1: $-3x + 2 - 2 < 11 - 2$ → $-3x < 9$
Step 2: $x > -3$ (sign reversed!)
Graph:
• Open circle at -3
• Shade right
• Solution: all numbers greater than -3
Solve:
Step 1: $-3x + 2 - 2 < 11 - 2$ → $-3x < 9$
Step 2: $x > -3$ (sign reversed!)
Graph:
• Open circle at -3
• Shade right
• Solution: all numbers greater than -3
10. Solve Advanced Linear Inequalities
Advanced Inequalities: Multi-step inequalities with variables on both sides, parentheses, or fractions
Steps for Advanced Inequalities:
Step 1: Simplify both sides (distribute, combine like terms)
Step 2: Move all variable terms to one side
Step 3: Move all constant terms to the other side
Step 4: Isolate the variable
Step 5: Check if sign needs reversing
Step 1: Simplify both sides (distribute, combine like terms)
Step 2: Move all variable terms to one side
Step 3: Move all constant terms to the other side
Step 4: Isolate the variable
Step 5: Check if sign needs reversing
Example 1: Solve $5x + 3 < 2x + 15$
Step 1: Subtract 2x from both sides
$5x - 2x + 3 < 15$
$3x + 3 < 15$
Step 2: Subtract 3
$3x < 12$
Step 3: Divide by 3
$x < 4$
Solution: $x < 4$
Step 1: Subtract 2x from both sides
$5x - 2x + 3 < 15$
$3x + 3 < 15$
Step 2: Subtract 3
$3x < 12$
Step 3: Divide by 3
$x < 4$
Solution: $x < 4$
Example 2: Solve $3(x - 2) \geq 2x + 5$
Step 1: Distribute
$3x - 6 \geq 2x + 5$
Step 2: Subtract 2x
$x - 6 \geq 5$
Step 3: Add 6
$x \geq 11$
Solution: $x \geq 11$
Step 1: Distribute
$3x - 6 \geq 2x + 5$
Step 2: Subtract 2x
$x - 6 \geq 5$
Step 3: Add 6
$x \geq 11$
Solution: $x \geq 11$
Example 3: Solve $-2(x + 4) > 3x - 7$
Step 1: Distribute
$-2x - 8 > 3x - 7$
Step 2: Subtract 3x
$-5x - 8 > -7$
Step 3: Add 8
$-5x > 1$
Step 4: Divide by -5 (REVERSE sign!)
$x < -\frac{1}{5}$
Solution: $x < -0.2$
Step 1: Distribute
$-2x - 8 > 3x - 7$
Step 2: Subtract 3x
$-5x - 8 > -7$
Step 3: Add 8
$-5x > 1$
Step 4: Divide by -5 (REVERSE sign!)
$x < -\frac{1}{5}$
Solution: $x < -0.2$
11. Graph Solutions to Advanced Linear Inequalities
Example: Solve and graph $4(x - 1) + 3 \leq 2x + 7$
Solve:
Step 1: Distribute: $4x - 4 + 3 \leq 2x + 7$
Step 2: Simplify: $4x - 1 \leq 2x + 7$
Step 3: Subtract 2x: $2x - 1 \leq 7$
Step 4: Add 1: $2x \leq 8$
Step 5: Divide by 2: $x \leq 4$
Graph: Closed circle at 4, shade left
Solve:
Step 1: Distribute: $4x - 4 + 3 \leq 2x + 7$
Step 2: Simplify: $4x - 1 \leq 2x + 7$
Step 3: Subtract 2x: $2x - 1 \leq 7$
Step 4: Add 1: $2x \leq 8$
Step 5: Divide by 2: $x \leq 4$
Graph: Closed circle at 4, shade left
12. Graph Compound Inequalities
Compound Inequality: Two inequalities joined by "AND" or "OR"
Conjunction (AND): Solution must satisfy BOTH inequalities
Disjunction (OR): Solution satisfies EITHER inequality (or both)
Conjunction (AND): Solution must satisfy BOTH inequalities
Disjunction (OR): Solution satisfies EITHER inequality (or both)
AND Compound Inequality:
• Form: $a < x < b$ or "$x > a$ AND $x < b$"
• Solution: Intersection (overlap) of both solution sets
• Graph: Shade the region between the two values
• Symbol: ∩ (intersection)
OR Compound Inequality:
• Form: "$x < a$ OR $x > b$"
• Solution: Union of both solution sets
• Graph: Shade both regions separately
• Symbol: ∪ (union)
• Form: $a < x < b$ or "$x > a$ AND $x < b$"
• Solution: Intersection (overlap) of both solution sets
• Graph: Shade the region between the two values
• Symbol: ∩ (intersection)
OR Compound Inequality:
• Form: "$x < a$ OR $x > b$"
• Solution: Union of both solution sets
• Graph: Shade both regions separately
• Symbol: ∪ (union)
AND Compound Inequalities (Conjunction)
Example 1: Graph $x > 2$ AND $x < 7$
(Can be written as $2 < x < 7$)
Graph:
• Open circle at 2
• Open circle at 7
• Shade BETWEEN 2 and 7
• Solution: all numbers between 2 and 7 (not including 2 or 7)
(Can be written as $2 < x < 7$)
Graph:
• Open circle at 2
• Open circle at 7
• Shade BETWEEN 2 and 7
• Solution: all numbers between 2 and 7 (not including 2 or 7)
Example 2: Graph $x \geq -3$ AND $x \leq 5$
(Written as $-3 \leq x \leq 5$)
Graph:
• Closed circle at -3
• Closed circle at 5
• Shade BETWEEN -3 and 5
• Solution: all numbers from -3 to 5 (including both endpoints)
(Written as $-3 \leq x \leq 5$)
Graph:
• Closed circle at -3
• Closed circle at 5
• Shade BETWEEN -3 and 5
• Solution: all numbers from -3 to 5 (including both endpoints)
OR Compound Inequalities (Disjunction)
Example 3: Graph $x < -2$ OR $x > 3$
Graph:
• Open circle at -2, shade LEFT
• Open circle at 3, shade RIGHT
• Two separate shaded regions (not connected)
• Solution: numbers less than -2 OR greater than 3
Graph:
• Open circle at -2, shade LEFT
• Open circle at 3, shade RIGHT
• Two separate shaded regions (not connected)
• Solution: numbers less than -2 OR greater than 3
Example 4: Graph $x \leq 1$ OR $x \geq 5$
Graph:
• Closed circle at 1, shade LEFT
• Closed circle at 5, shade RIGHT
• Two separate shaded regions
• Solution: numbers ≤ 1 OR numbers ≥ 5
Graph:
• Closed circle at 1, shade LEFT
• Closed circle at 5, shade RIGHT
• Two separate shaded regions
• Solution: numbers ≤ 1 OR numbers ≥ 5
| Type | Keyword | Graph Style | Example |
|---|---|---|---|
| Conjunction | AND | Shade between (overlap) | $-2 < x < 5$ |
| Disjunction | OR | Shade both regions separately | $x < -2$ OR $x > 5$ |
13. Write Compound Inequalities from Graphs
Steps to Write Compound Inequalities:
Step 1: Identify the critical points (where circles are)
Step 2: Determine circle types (open or closed)
Step 3: Identify shading pattern:
• Shading between two points → AND
• Shading in two separate regions → OR
Step 4: Write the appropriate inequality
Step 1: Identify the critical points (where circles are)
Step 2: Determine circle types (open or closed)
Step 3: Identify shading pattern:
• Shading between two points → AND
• Shading in two separate regions → OR
Step 4: Write the appropriate inequality
Example 1: A graph shows closed circles at -1 and 4 with shading between them.
Analysis:
• Closed circles → include equal signs
• Shading between → AND
• Answer: $-1 \leq x \leq 4$ or "$x \geq -1$ AND $x \leq 4$"
Analysis:
• Closed circles → include equal signs
• Shading between → AND
• Answer: $-1 \leq x \leq 4$ or "$x \geq -1$ AND $x \leq 4$"
Example 2: A graph shows an open circle at 0 shading left, and an open circle at 6 shading right.
Analysis:
• Open circles → no equal signs
• Two separate regions → OR
• Answer: $x < 0$ OR $x > 6$
Analysis:
• Open circles → no equal signs
• Two separate regions → OR
• Answer: $x < 0$ OR $x > 6$
Example 3: A graph shows closed circle at -5 and open circle at 2 with shading between.
Analysis:
• Closed at -5, open at 2
• Shading between → AND
• Answer: $-5 \leq x < 2$
Analysis:
• Closed at -5, open at 2
• Shading between → AND
• Answer: $-5 \leq x < 2$
14. Solve Compound Inequalities
Solving AND Compound Inequalities
Method 1: Solve Together (when in form $a < x < b$)
• Perform the same operation on ALL three parts
• Keep inequalities in order
Method 2: Solve Separately
• Break into two inequalities
• Solve each independently
• Combine using AND (find overlap)
• Perform the same operation on ALL three parts
• Keep inequalities in order
Method 2: Solve Separately
• Break into two inequalities
• Solve each independently
• Combine using AND (find overlap)
Example 1: Solve $-3 < 2x + 1 < 9$
Method: Solve all at once
Step 1: Subtract 1 from all parts
$-3 - 1 < 2x + 1 - 1 < 9 - 1$
$-4 < 2x < 8$
Step 2: Divide all parts by 2
$\frac{-4}{2} < \frac{2x}{2} < \frac{8}{2}$
$-2 < x < 4$
Solution: $-2 < x < 4$
Method: Solve all at once
Step 1: Subtract 1 from all parts
$-3 - 1 < 2x + 1 - 1 < 9 - 1$
$-4 < 2x < 8$
Step 2: Divide all parts by 2
$\frac{-4}{2} < \frac{2x}{2} < \frac{8}{2}$
$-2 < x < 4$
Solution: $-2 < x < 4$
Example 2: Solve $x + 5 > 3$ AND $2x \leq 10$
Solve first inequality:
$x + 5 > 3$
$x > -2$
Solve second inequality:
$2x \leq 10$
$x \leq 5$
Combine with AND (overlap):
$x > -2$ AND $x \leq 5$
Solution: $-2 < x \leq 5$
Solve first inequality:
$x + 5 > 3$
$x > -2$
Solve second inequality:
$2x \leq 10$
$x \leq 5$
Combine with AND (overlap):
$x > -2$ AND $x \leq 5$
Solution: $-2 < x \leq 5$
Solving OR Compound Inequalities
Strategy:
• Solve each inequality separately
• Write both solutions with OR between them
• Solutions remain separate (don't combine)
• Solve each inequality separately
• Write both solutions with OR between them
• Solutions remain separate (don't combine)
Example 3: Solve $3x - 5 < 4$ OR $2x + 1 > 11$
Solve first inequality:
$3x - 5 < 4$
$3x < 9$
$x < 3$
Solve second inequality:
$2x + 1 > 11$
$2x > 10$
$x > 5$
Solution: $x < 3$ OR $x > 5$
Solve first inequality:
$3x - 5 < 4$
$3x < 9$
$x < 3$
Solve second inequality:
$2x + 1 > 11$
$2x > 10$
$x > 5$
Solution: $x < 3$ OR $x > 5$
Example 4: Solve $-2x + 3 \geq 7$ OR $x - 4 > 1$
Solve first inequality:
$-2x + 3 \geq 7$
$-2x \geq 4$
$x \leq -2$ (sign reversed)
Solve second inequality:
$x - 4 > 1$
$x > 5$
Solution: $x \leq -2$ OR $x > 5$
Solve first inequality:
$-2x + 3 \geq 7$
$-2x \geq 4$
$x \leq -2$ (sign reversed)
Solve second inequality:
$x - 4 > 1$
$x > 5$
Solution: $x \leq -2$ OR $x > 5$
15. Graph Solutions to Compound Inequalities
Example 1 (AND): Solve and graph $-4 \leq 3x - 1 < 8$
Solve:
Add 1 to all parts: $-3 \leq 3x < 9$
Divide by 3: $-1 \leq x < 3$
Graph:
• Closed circle at -1
• Open circle at 3
• Shade between -1 and 3
• Solution: $-1 \leq x < 3$
Solve:
Add 1 to all parts: $-3 \leq 3x < 9$
Divide by 3: $-1 \leq x < 3$
Graph:
• Closed circle at -1
• Open circle at 3
• Shade between -1 and 3
• Solution: $-1 \leq x < 3$
Example 2 (OR): Solve and graph $2x + 5 < 1$ OR $x - 3 \geq 2$
Solve first:
$2x + 5 < 1$
$2x < -4$
$x < -2$
Solve second:
$x - 3 \geq 2$
$x \geq 5$
Graph:
• Open circle at -2, shade left
• Closed circle at 5, shade right
• Two separate regions
• Solution: $x < -2$ OR $x \geq 5$
Solve first:
$2x + 5 < 1$
$2x < -4$
$x < -2$
Solve second:
$x - 3 \geq 2$
$x \geq 5$
Graph:
• Open circle at -2, shade left
• Closed circle at 5, shade right
• Two separate regions
• Solution: $x < -2$ OR $x \geq 5$
Special Cases:
• No Solution: If AND inequality has no overlap (e.g., $x < -5$ AND $x > 3$)
• All Real Numbers: If OR inequality covers everything (e.g., $x < 5$ OR $x > 0$)
• No Solution: If AND inequality has no overlap (e.g., $x < -5$ AND $x > 3$)
• All Real Numbers: If OR inequality covers everything (e.g., $x < 5$ OR $x > 0$)
Quick Reference Guide
Inequality Symbols Quick Reference:
• $<$ : less than (open circle, shade left)
• $>$ : greater than (open circle, shade right)
• $\leq$ : less than or equal (closed circle, shade left)
• $\geq$ : greater than or equal (closed circle, shade right)
• $<$ : less than (open circle, shade left)
• $>$ : greater than (open circle, shade right)
• $\leq$ : less than or equal (closed circle, shade left)
• $\geq$ : greater than or equal (closed circle, shade right)
⚠️ GOLDEN RULE:
REVERSE THE INEQUALITY SIGN when:
• Multiplying both sides by a negative number
• Dividing both sides by a negative number
DO NOT reverse when:
• Adding or subtracting any number
• Multiplying or dividing by a positive number
REVERSE THE INEQUALITY SIGN when:
• Multiplying both sides by a negative number
• Dividing both sides by a negative number
DO NOT reverse when:
• Adding or subtracting any number
• Multiplying or dividing by a positive number
Solving Steps Summary:
One-Step:
• Use inverse operation
• Watch for negative multiplier/divisor
Two-Step:
1. Add/subtract to isolate variable term
2. Multiply/divide to isolate variable
Multi-Step:
1. Distribute and combine like terms
2. Move variables to one side
3. Move constants to other side
4. Isolate variable
One-Step:
• Use inverse operation
• Watch for negative multiplier/divisor
Two-Step:
1. Add/subtract to isolate variable term
2. Multiply/divide to isolate variable
Multi-Step:
1. Distribute and combine like terms
2. Move variables to one side
3. Move constants to other side
4. Isolate variable
Compound Inequalities:
AND (Conjunction):
• Written as: $a < x < b$ or "$x > a$ AND $x < b$"
• Graph: Shade between the values
• Solution: Intersection (overlap)
OR (Disjunction):
• Written as: "$x < a$ OR $x > b$"
• Graph: Shade two separate regions
• Solution: Union (either or both)
AND (Conjunction):
• Written as: $a < x < b$ or "$x > a$ AND $x < b$"
• Graph: Shade between the values
• Solution: Intersection (overlap)
OR (Disjunction):
• Written as: "$x < a$ OR $x > b$"
• Graph: Shade two separate regions
• Solution: Union (either or both)
Graphing Checklist:
✓ Identify critical value(s)
✓ Choose correct circle type (open vs closed)
✓ Shade in correct direction(s)
✓ Draw arrows to show continuation
✓ For compound: check if AND (between) or OR (separate)
✓ Identify critical value(s)
✓ Choose correct circle type (open vs closed)
✓ Shade in correct direction(s)
✓ Draw arrows to show continuation
✓ For compound: check if AND (between) or OR (separate)
| Inequality Type | Example | Solution Process |
|---|---|---|
| One-Step | $x + 5 < 9$ | Subtract 5 → $x < 4$ |
| One-Step (negative) | $-3x > 12$ | Divide by -3, flip → $x < -4$ |
| Two-Step | $2x + 3 \leq 11$ | Subtract 3, divide by 2 → $x \leq 4$ |
| Multi-Step | $3(x-2) > x+4$ | Distribute, combine, solve → $x > 5$ |
| Compound (AND) | $-3 < x < 5$ | Shade between -3 and 5 |
| Compound (OR) | $x<-2$ OR $x>3$ | Shade two regions |
Success Tips for Solving Inequalities:
✓ Treat inequalities like equations EXCEPT when multiplying/dividing by negatives
✓ Always check your solution by testing a value
✓ Write your final answer clearly with proper symbols
✓ Draw number line graphs carefully with correct circles
✓ For compound inequalities, identify AND vs OR first
✓ Practice reversing signs with negative operations
✓ Show all work step-by-step
✓ Treat inequalities like equations EXCEPT when multiplying/dividing by negatives
✓ Always check your solution by testing a value
✓ Write your final answer clearly with proper symbols
✓ Draw number line graphs carefully with correct circles
✓ For compound inequalities, identify AND vs OR first
✓ Practice reversing signs with negative operations
✓ Show all work step-by-step