5.3 Linear Inequalities in One Variable with Applications

Linear inequalities help us compare quantities and find ranges of solutions. They're like equations, but with greater-than or less-than symbols instead of equals signs. We use them to set limits or boundaries in math problems.

Solving inequalities involves similar steps to solving equations, but with a few twists. We can graph them on number lines, use interval notation, and apply them to real-world situations. Understanding inequalities is key to tackling many practical math problems.

Linear Inequalities in One Variable

Graphing linear inequalities

• Inequalities compare two quantities using symbols
  • $<$ represents "less than" (5 < 7)
  • $>$ represents "greater than" (9 > 2)
  • $\leq$ represents "less than or equal to" (x ≤ 4)
  • $\geq$ represents "greater than or equal to" (y ≥ -1)
• Graphing inequalities on a number line involves
  • Using an open circle (○) for strict inequalities ($<$ or $>$)
  • Using a closed circle (●) for inclusive inequalities ($\leq$ or $\geq$)
  • Shading the portion of the number line that satisfies the inequality
    • Shade to the right for $>$ or $\geq$ (x > 3 shades values greater than 3)
    • Shade to the left for $<$ or $\leq$ (y ≤ 0 shades values less than or equal to 0)
  • The shaded portion represents the range of the inequality
• Interval notation expresses the solution set of an inequality concisely
  • Use parentheses ( ) for strict inequalities (x < 5 is written as (-∞, 5))
  • Use brackets [ ] for inclusive inequalities (x ≥ -2 is written as [-2, ∞))
  • $-\infty$ represents negative infinity and $\infty$ represents positive infinity
  • $x > 3$ is written as $(3, \infty)$ in interval notation

Algebraic solutions for inequalities

• Solving linear inequalities algebraically is similar to solving linear equations
  • Perform the same operation on both sides of the inequality to maintain the inequality (if x + 3 < 7, then x < 4)
• When multiplying or dividing by a negative number, reverse the inequality symbol
  • If $-2x < 6$, then $x > -3$ (dividing by -2 flips the inequality)
• Isolate the variable on one side of the inequality by
  • Adding or subtracting the same value from both sides (x - 5 ≥ 1 becomes x ≥ 6)
  • Multiplying or dividing both sides by the same positive value (2x < 10 becomes x < 5)
• Check your solution by substituting a value from the solution set into the original inequality (if x > 2, then 3 > 2 is true)
• The solution set represents the domain of the inequality

Linear Functions and Inequalities

• A linear function is represented by an equation in the form y = mx + b
  • m represents the slope of the line
  • b represents the y-intercept (where the line crosses the y-axis)
• Linear inequalities are closely related to linear functions
  • They use the same symbols (>, <, ≥, ≤) to compare a linear expression to a constant or another linear expression
• A system of inequalities involves two or more inequalities that must be satisfied simultaneously
• A compound inequality combines two inequalities using "and" or "or"
  • For example, 2 < x < 5 means x is greater than 2 and less than 5

Real-world applications of inequalities

• Identify the unknown quantity and assign a variable (let x represent the number of tickets sold)
• Write an inequality that represents the constraints or limits in the problem
  • Translate verbal phrases into mathematical symbols
    • "At least" means $\geq$ (the store needs to sell at least 50 items)
    • "At most" means $\leq$ (the truck can carry at most 2,000 pounds)
    • "More than" means $>$ (the company needs to produce more than 500 units)
    • "Less than" means $<$ (the project must be completed in less than 30 days)
• Solve the inequality algebraically (if 50 ≤ 2x + 10, then x ≥ 20)
• Interpret the solution in the context of the problem
  • Check if the solution makes sense given the constraints (the company must produce at least 20 items to meet the minimum requirement)
• Example: A store requires a minimum purchase of $$50 to qualify for free shipping. If sales tax is 6%, what is the minimum amount before tax needed to get free shipping?
  1. Let $x$ represent the amount spent before tax
  2. The total cost with tax must be at least $50:$1.06x \geq 50$$
  3. Solve the inequality: $x \geq \frac{50}{1.06} \approx 47.17$
  4. Interpret the result: You need to spend at least $$47.17 before tax to qualify for free shipping