2.5 Solve Linear Inequalities

Linear inequalities let you describe a range of values that satisfy a condition, rather than pinpointing a single answer like equations do. They show up constantly in real-world situations where you're dealing with limits, minimums, or budgets. This section covers how to graph them, solve them, and translate word problems into inequality form.

Graphing and Solving Linear Inequalities

Graphing inequalities on number lines

Inequalities compare quantities using four symbols: $<$, $>$, $\leq$, and $\geq$. Unlike equations, their solutions aren't a single point. Instead, you get a whole set of values, and a number line is the best way to show that visually.

Two things determine how you draw the graph: the circle type and the shading direction.

• Open circle ○ means the endpoint is not included (used with $<$ or $>$)
• Closed circle ● means the endpoint is included (used with $\leq$ or $\geq$)

For shading direction:

• $<$ or $\leq$: shade to the left of the endpoint (values getting smaller)
• $>$ or $\geq$: shade to the right of the endpoint (values getting larger)

Examples:

• $x < 3$: open circle at 3, shade left. This means every number less than 3 works, but 3 itself does not.
• $y \geq -2$: closed circle at $-2$, shade right. Here $-2$ is included, along with everything greater.

Solving linear inequalities

You solve linear inequalities almost exactly like linear equations, with one critical difference: when you multiply or divide both sides by a negative number, you must flip the inequality sign. This is the single most common mistake students make, so watch for it every time.

Steps for solving:

1. Simplify each side if needed (distribute, combine like terms).
2. Add or subtract to move constants to one side and variable terms to the other.
3. Multiply or divide to isolate the variable. If you multiply or divide by a negative, reverse the inequality symbol.
4. Write the solution and graph it or express it in interval notation.

Example 1: Solve $2x - 5 < 7$

1. Add 5 to both sides: $2x < 12$
2. Divide both sides by 2: $x < 6$
3. Solution: all values less than 6. In interval notation: $(-\infty, 6)$

Example 2 (variable on both sides): Solve $3x + 2 \geq 4x - 1$

1. Subtract $3x$ from both sides: $2 \geq x - 1$

2. Add 1 to both sides: $3 \geq x$, which is the same as $x \leq 3$

3. Solution: $(-\infty, 3]$

Example 3 (negative division): Solve $-4x + 8 > 20$

1. Subtract 8 from both sides: $-4x > 12$
2. Divide both sides by $-4$ and flip the sign: $x < -3$
3. Solution: $(-\infty, -3)$

That third example is the type to practice repeatedly. The sign flip only happens when you multiply or divide by a negative, not when you add or subtract one.

Word problems to inequalities

Translating word problems into inequalities takes practice, but the process is consistent:

1. Identify the unknown and assign it a variable ($x$).

2. Translate key phrases into math symbols:

   • "at least" → $\geq$
   • "at most" → $\leq$
   • "more than" → $>$
   • "less than" → $<$
   • "no more than" → $\leq$

3. Build the inequality from the relationships described.

4. Solve using the methods above.

5. Interpret your answer back in the context of the problem.

Example: John needs at least $50 more to buy a game console costing $250. How much money does he currently have?

• Let $x$ = the amount John currently has.
• "Needs at least $50 more" means his money plus $50 must reach $250: $x + 50 \geq 250$
• Subtract 50: $x \geq 200$
• John currently has at least $200.

Pay attention to the phrase direction. "At least 50 more" tells you the gap between what he has and what he needs is at least 50, which means his current amount could be $200 or less. Read carefully and check that your inequality makes sense before solving.

Real-world applications of inequalities

Many real situations involve constraints rather than exact values. Inequalities are the natural tool for modeling these.

Common scenarios:

• Budget limits: Total spending must stay at or below a set amount. If you're planning an event with a $5,000 budget, spending $x$ on a venue and $2,000 on catering gives you $x + 2000 \leq 5000$, so $x \leq 3000$.
• Minimum requirements: A store must order at least 100 units to qualify for a bulk discount. If they've already ordered 65, they need $65 + x \geq 100$, meaning $x \geq 35$ more units.
• Acceptable ranges: A manufacturer needs a part's length to be no more than 0.5 cm from 10 cm. That's two inequalities: $9.5 \leq x \leq 10.5$.

When interpreting solutions in context, always consider practical limits. If your inequality gives $x \geq 35$ and $x$ represents units ordered, only whole numbers make sense. If $x$ represents money, negative values probably don't apply.

Additional Techniques and Notation

Interval notation gives you a compact way to write solution sets:

When solving more complex inequalities, you can always check your answer by plugging in a value from your solution set. If you got $x < 6$, try $x = 0$ in the original inequality. If it works, you're likely on the right track. If you try a value outside the set (like $x = 7$) and it also works, something went wrong.