3.6 Solve Applications with Linear Inequalities

Translating and Solving Linear Inequality Applications

Many real-world problems don't have a single answer. Instead, they have a range of acceptable values: you need to earn at least a certain score, spend no more than a budget, or stay under a weight limit. Linear inequalities let you model these constraints mathematically and find every value that works.

Translating Real-World Scenarios

The hardest part of these problems is turning English into math. Here's a reliable process:

1. Read the problem and identify the unknown. Assign it a variable (like $x$).
2. Spot the constraint language. These key phrases tell you which inequality symbol to use:

1. Write the inequality. Build an expression on one side using the variable, and place the constraint value on the other side with the correct symbol.
2. Check for compound inequalities. Some problems have two constraints at once. For example, "between 18 and 65 years old" translates to $18 \leq x \leq 65$.

Solving Practical Inequalities

Solving a linear inequality works almost exactly like solving a linear equation, with one critical difference.

1. Add or subtract the same value from both sides to move constants away from the variable.
2. Multiply or divide both sides by the same positive value to isolate the variable.
3. If you multiply or divide by a negative number, flip the inequality symbol. This is the rule students forget most often.

Example: A student needs to spend no more than $200 on books. She has already spent $75 and each remaining book costs $25. How many more books can she buy?

• Set up: $75 + 25x \leq 200$
• Subtract 75: $25x \leq 125$
• Divide by 25: $x \leq 5$

She can buy at most 5 more books.

Example with a negative coefficient: Solve $-3x + 12 > 0$

• Subtract 12: $-3x > -12$
• Divide by $-3$ and flip the symbol: $x < 4$

Graphing and Writing Solution Sets

Once you've solved the inequality, represent the solution on a number line:

• Open circle ($\circ$) for strict inequalities ($<$ or $>$): the endpoint is not included.
• Closed circle ($\bullet$) for inclusive inequalities ($\leq$ or $\geq$): the endpoint is included.
• Shade in the direction of all values that satisfy the inequality.

Then express the solution in interval notation:

• $x \leq 5$ → $(-\infty, 5]$
• $x > 4$ → $(4, \infty)$
• $18 \leq x \leq 65$ → $[18, 65]$

Square brackets mean the endpoint is included; parentheses mean it's not. Infinity always gets a parenthesis because you can never actually reach it.

Interpreting Inequality Solutions

Finding the mathematical answer is only half the job. You also need to make sure it fits the real-world situation.

• Translate back to context. Don't just write $x \leq 5$. Say: "She can buy 5 or fewer additional books."
• Check for reasonableness. If your variable represents a number of people, negative or fractional answers don't make sense. You may need to round or restrict the solution further.
• Verify units. Your answer should use the same units the problem asked about (dollars, hours, tickets, etc.).
• Test a value. Pick a number from your solution set and plug it back into the original inequality to confirm it works. For the book example, try $x = 3$: $75 + 25(3) = 150$, and $150 \leq 200$ ✓.