📊 Linear Inequalities

1. Comparison Relations & Verbal Meanings

• Strictly Less Than ($<$): The value is strictly smaller than the boundary. Verbal cues: under, less than, below.
• Strictly Greater Than ($>$): The value is strictly larger than the boundary. Verbal cues: more than, greater than, exceeds, above.
• Less Than or Equal To ($\le$): The value is at most the boundary. Verbal cues: at most, maximum of, no more than.
• Greater Than or Equal To ($\ge$): The value is at least the boundary. Verbal cues: at least, minimum of, no less than.
• Not Equal To ($\neq$): Any real number except the boundary point.

2. Number Line Graphing Rules

• Boundary Point Circle Type:
  - Open Circle ($\circ$, Hollow): Used for strict inequalities ($<$ and $>$). The boundary value itself is not part of the solution.
  - Solid Circle ($\bullet$, Filled): Used for inclusive inequalities ($\le$ and $\ge$). The boundary value is part of the solution.
  
• Ray Shading Direction:
  - Rightward Shading ($>$ or $\ge$): Shading extends to the right towards positive infinity ($+\infty$).
  - Leftward Shading ($<$ or $\le$): Shading extends to the left towards negative infinity ($-\infty$).
  - Not Equal To ($\neq$): An open circle ($\circ$) is placed at the boundary point, and the line is shaded in both directions ($< b$ and $> b$).

• Step 1 — Identify the Boundary Point: Locate the numerical value where the condition changes.
• Step 2 — Determine Circle Style: Inspect the inequality symbol. Choose an open circle ($\circ$) for strict ($<$, $>$) or a solid circle ($\bullet$) for inclusive ($\le$, $\ge$).
• Step 3 — Determine Ray Direction: Point the arrow right for larger values ($>$ or $\ge$) or left for smaller values ($<$ or $\le$).
• Step 4 — Verify with a Test Point: Select a simple test value (such as $x = 0$) in the shaded region and verify that substituting it makes the inequality true.

1. Addition and Subtraction Properties of Inequality

• If $a < b$, then $a + c < b + c$ and $a - c < b - c$
• If $a \le b$, then $a + c \le b + c$ and $a - c \le b - c$
• If $a > b$, then $a + c > b + c$ and $a - c > b - c$
• If $a \ge b$, then $a + c \ge b + c$ and $a - c \ge b - c$
  
• Why this works: It allows us to undo addition with subtraction and undo subtraction with addition, leaving the variable isolated while preserving the inequality.

2. Step-by-Step Solver Guide

• Step 1 — Identify the Operation: Determine which operation and number are affecting the variable ($x$).
• Step 2 — Apply the Inverse Operation: Perform the opposite operation on both sides of the inequality to isolate $x$.
• Step 3 — Simplify: Calculate the simplified result. Remember: the direction of the inequality sign remains unchanged!
• Step 4 — Graph the Solution: Mark the boundary point with an open circle (for $<$ or $>$) or a solid circle (for $\le$ or $\ge$), then shade the corresponding ray.

1. Multiplication & Division Properties of Inequalities

• Multiplying/Dividing by a Positive Number ($c > 0$): The inequality direction remains unchanged:
  
  $$\text{If } a < b \text{ and } c > 0 \implies a \cdot c < b \cdot c \quad \text{and} \quad \frac{a}{c} < \frac{b}{c}$$
  
  Example: $2x < 10 \implies x < 5$
  
• ⚠️ The Golden Rule — Multiplying/Dividing by a Negative Number ($c < 0$): When multiplying or dividing both sides by a negative number, the inequality sign MUST FLIP direction:
  
  $$\text{If } a < b \text{ and } c < 0 \implies a \cdot c > b \cdot c \quad \text{and} \quad \frac{a}{c} > \frac{b}{c}$$
• Why does the sign flip? (Two Intuitive Explanations):
  1. Geometric / Number Line Reason: Consider $2 < 5$. If we multiply both sides by $-1$, we get $-2$ and $-5$. On the real number line, $-2$ lies to the right of $-5$, so $-2 > -5$. Multiplying or dividing by a negative number reflects positions across zero, reversing the relative order.
  2. Algebraic Explanation by Moving Both Sides: Consider $-x > -3$. If we add $+x$ to both sides, we get $0 > -3 + x$. Next, adding $+3$ to both sides gives $3 > x$, which directly means $x < 3$. Thus, $-x > -3 \iff x < 3$. This is exactly the same result as multiplying both sides by $-1$, which reverses the inequality. In this sense, moving both sides to the opposite side is algebraically equivalent to multiplying the entire inequality by $-1$.

• $3x \le -12 \implies x \le -4$ (Divided by $+3 > 0$, sign stays $\le$)
• $-3x \le 12 \implies x \ge -4$ (Divided by $-3 < 0$, sign flips to $\ge$)

3. Step-by-Step Solver Method

• Step 1 — Identify the Coefficient & Operation: Determine the number and operation affecting the variable.
• Step 2 — Apply the Inverse Operation: Divide if the variable is multiplied, or multiply if the variable is divided.
• Step 3 — 🚨 Sign Flip Check: If the multiplier/divisor is negative ($c < 0$), reverse the inequality sign ($< \longleftrightarrow >$ and $\le \longleftrightarrow \ge$). If positive ($c > 0$), keep the sign unchanged.
• Step 4 — Simplify & Graph: Calculate the numerical boundary point, select open/solid circle, and shade the solution ray.

1. The Sequential Two-Step Strategy

• Step 1 — Isolate the Variable Term: Add or subtract constant terms from both sides to gather all constants on the opposite side.
  
  $$ax + b < c \implies ax < c - b$$
• Step 2 — Isolate the Variable: Multiply or divide both sides by the coefficient of the variable. 🚨 Golden Rule: If dividing or multiplying by a negative coefficient, remember to flip the inequality symbol!

• Expanding Parentheses (Distributive Property): Multiply the outside term into every term inside the parentheses:
  
  $$a(x + b) \le c \implies ax + ab \le c$$
  
  Example: $3(x - 2) \le 15 \implies 3x - 6 \le 15 \implies 3x \le 21 \implies x \le 7$
  
• Combining Like Terms on One Side: Add or subtract the coefficients of identical variable terms before isolating:
  Example: $5x - 2x + 4 > 19 \implies 3x + 4 > 19 \implies 3x > 15 \implies x > 5$

3. Step-by-Step Solver Method

• Step 1 — Simplify Expressions: Expand parentheses or combine like terms.
• Step 2 — Isolate the Variable Term: Add or subtract constants from both sides.
• Step 3 — Isolate the Variable: Divide or multiply by the coefficient (flip the sign if negative!).
• Step 4 — Graph & Verify: Place an open/closed circle at the boundary value, shade the solution ray, and test a point.

1. Bounded Intervals & Compound "AND" Inequalities

• Form: $a < x \le b$ means $x$ is simultaneously strictly greater than $a$ and less than or equal to $b$ ($x > a \text{ and } x \le b$).
• Verbal cues: between $a$ and $b$, from $a$ to $b$, bounded by $a$ and $b$.
• Solution Set: All real numbers that satisfy both boundary conditions.

• Left Boundary ($a$): Place an open circle ($\circ$) if $a < x$ or a solid circle ($\bullet$) if $a \le x$.
• Right Boundary ($b$): Place an open circle ($\circ$) if $x < b$ or a solid circle ($\bullet$) if $x \le b$.
• Shading: Shade the continuous segment strictly between $a$ and $b$.

3. Solving Double-Sided Inequalities: Simultaneous 3-Part Operations

• Step 1 — Isolate the variable term: Subtract or add constant $c$ to all 3 parts:
  
  $$a - c \le mx < b - c$$
• Step 2 — Isolate the variable: Divide all 3 parts by $m$.
  🚨 Sign Reversal: If $m < 0$, reverse both inequality signs and flip the interval bounds!

1. Modeling Real-World Contexts with Inequalities

• Cap / Maximum Limit ($\le$): Spending within a budget, elevator weight capacity, battery life.
• Goal / Minimum Target ($\ge$): Fundraising goals, minimum score requirements, savings targets.
• Strict Ceilings & Floors ($<$ or $>$): Strict speed limits, weight restrictions without exceeding boundaries.

• At most, maximum of, no more than, does not exceed: $\le$
• At least, minimum of, no less than, must reach: $\ge$
• Fewer than, under, below, strictly less: $<$
• More than, exceeds, above, strictly greater: $>$

3. The 4-Step Word Problem Framework

• Step 1 — Define Variable: Identify the unknown quantity and assign a variable (e.g., Let $b$ = number of books).
• Step 2 — Set Up Inequality: Combine unit rate and fixed quantities with the appropriate inequality symbol.
• Step 3 — Solve Algebraically: Apply inverse operations to isolate the variable.
• Step 4 — Interpret Contextually: If the item cannot be split into fractions (e.g. books, people, boxes, weeks), round down for maximum quantities and round up for minimum quantities.