Linear inequalities are a powerful tool for comparing values and finding ranges of solutions. They use symbols like < and > to show relationships between numbers, and can be graphed on number lines using open or closed circles.

Solving linear inequalities involves similar steps to solving equations, but with a twist. When multiplying or dividing by negative numbers, you flip the inequality symbol. This concept is crucial for tackling more complex problems and word problems.

Linear Inequalities

Linear inequalities on number lines

• Represent relationships between two values using symbols $<$ (less than), $>$ (greater than), $\leq$ (less than or equal to), and $\geq$ (greater than or equal to)
• Graph inequalities on number lines using open circles (○) for strict inequalities ($<$ or $>$) and closed circles (●) for inclusive inequalities ($\leq$ or $\geq$)
  • Shade number line to left of 5 for $x < 5$ with open circle at 5
  • Shade number line to right of -2 for $x \geq -2$ with closed circle at -2

Addition and subtraction in inequalities

• Solve linear inequalities by performing same operation on both sides
• Inequality symbol remains unchanged when adding or subtracting same value from both sides
• Solve $x - 3 < 7$ by adding 3 to both sides: $x - 3 + 3 < 7 + 3$, simplifying to $x < 10$
• Use inverse operations to isolate the variable on one side of the inequality

Multiplication and division in inequalities

• Solve linear inequalities by performing same operation on both sides
• Reverse direction of inequality symbol when multiplying or dividing both sides by negative number
• Solve $-2x > 6$ by dividing both sides by -2: $\frac{-2x}{-2} < \frac{6}{-2}$, simplifying to $x < -3$ (symbol changes from $>$ to $<$)

Complex linear inequalities

• Combine like terms on each side of inequality symbol to simplify
• Isolate variable by performing same operations on both sides
• Solve $3x - 5 + 2x \geq 4x - 7$ by combining like terms: $5x - 5 \geq 4x - 7$
  • Subtract 4x from both sides: $x - 5 \geq -7$
  • Add 5 to both sides: $x \geq -2$

Word problems to inequalities

• Represent unknown quantity with variable ($x$)
• Translate word problem into linear inequality using given information
• Solve resulting linear inequality
• Example: Sum of three times number and 5 is at most 20. Find range of possible values.
  1. Let $x$ represent unknown number
  2. Translate word problem: $3x + 5 \leq 20$
  3. Solve inequality:
     • Subtract 5 from both sides: $3x \leq 15$
     • Divide both sides by 3: $x \leq 5$
  4. Range of possible values is $x \leq 5$ (number is less than or equal to 5)

Algebraic Expressions and Graphing

• Linear inequalities often involve algebraic expressions with variables
• Solve inequalities by manipulating these expressions using mathematical operations
• After solving, represent the solution by graphing on a number line or coordinate plane
• Graphing helps visualize the range of values that satisfy the inequality