2.5 Solve Linear Inequalities

Linear inequalities expand on equations, allowing us to compare values using symbols like < and >. We'll learn to graph these on number lines, using open or closed circles to show which values are included in the solution set.

Solving linear inequalities is similar to equations, but with a twist: flip the sign when multiplying or dividing by negatives. We'll practice turning word problems into inequalities and explore real-world applications, from budgets to product orders.

Graphing and Solving Linear Inequalities

Graphing inequalities on number lines

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• Inequalities compare quantities using inequality symbols ($<$, $>$, $\leq$, and $\geq$)
• Graph inequalities on a number line using an open circle ○ for strict inequalities ($<$ or $>$) and a closed circle ● for inclusive inequalities ($\leq$ or $\geq$)
  • Open circle ○ excludes the endpoint from the solution set
  • Closed circle ● includes the endpoint in the solution set
• Shade the portion of the number line satisfying the inequality
  • For $<$ or $\leq$, shade left of the endpoint
  • For $>$ or $\geq$, shade right of the endpoint
• Examples:
  • $x < 3$ is graphed with an open circle ○ at 3 and shading to the left
  • $y \geq -2$ is graphed with a closed circle ● at -2 and shading to the right

Solving linear inequalities

• Solve linear inequalities using similar steps as solving linear equations, but with one key rule:
  • When multiplying or dividing both sides by a negative number, reverse the inequality sign
• Isolate the variable by performing the same operation on both sides
  • Add or subtract the same value to eliminate constants
  • Multiply or divide by the same non-zero value to eliminate coefficients
• For inequalities with variables on both sides:
  1. Move all terms with the variable to one side
  2. Move all constants to the other side
  3. Combine like terms as needed
• Simplify the inequality to isolate the variable on one side
• Graph the solution set on a number line or express it using interval notation
• Examples:
  • $2x - 5 < 7$ becomes $2x < 12$ after adding 5 to both sides, then $x < 6$ after dividing by 2
  • $3x + 2 \geq 4x - 1$ becomes $2 \geq x - 1$ after subtracting $3x$ from both sides, then $3 \geq x$ after adding 1 to both sides

Word problems to inequalities

• Identify the unknown quantity and assign it a variable ($x$)
• Translate the given information into an inequality using the variable
  • Use inequality symbols for phrases like "at least" ($\geq$), "at most" ($\leq$), "more than" ($>$), or "less than" ($<$)
  • Carefully consider the direction of the inequality sign when translating
• Solve the resulting inequality using the methods described above
• Interpret the solution in the context of the original word problem
• Example:
  • "John needs at least $50 more to buy a game console costing$250. How much money does he have?"
  • Let $x$ be the amount of money John has
  • Translate to inequality: $x + 50 \geq 250$
  • Solve: $x \geq 200$, so John has at least $200

Real-world applications of inequalities

• Recognize situations where linear inequalities model constraints or requirements (budget limitations, min/max quantities, acceptable ranges)
• Create an inequality representing the constraints
  • Define variables and their meanings
  • Identify coefficients and constants based on given information
• Solve the inequality to find the solution set
• Interpret the solution in terms of the original context
  • Determine the meaning for the quantities involved
  • Consider practical limitations or considerations affecting interpretation
• Examples:
  • A store must order at least 100 units of a product to qualify for a discount
  • A project's budget cannot exceed $5000 while allocating at least$1000 for materials and $2000 for labor

Additional Techniques and Notation

• Graphing techniques: Use number lines to visually represent solutions of linear inequalities
• Interval notation: Express solutions in a compact form, e.g., $x > 3$ as $(3,\infty)$
• Algebraic manipulation: Apply operations to both sides of an inequality to isolate the variable
• Variable isolation: Rearrange terms to get the variable alone on one side of the inequality