Inequalities

Hey students! 👋 Welcome to one of the most powerful tools in mathematics - inequalities! Today we're going to explore how to solve both linear and quadratic inequalities, represent them visually on number lines, and express solutions using interval notation. By the end of this lesson, you'll be confidently tackling inequality problems and understanding exactly what those solution sets mean in real-world contexts. Think of inequalities as mathematical statements that help us describe ranges of values - like determining the safe speed limits on roads or calculating profit margins in business! 🚗💰

Understanding Linear Inequalities

Linear inequalities are similar to linear equations, but instead of an equals sign, we use inequality symbols: < (less than), > (greater than), ≤ (less than or equal to), and ≥ (greater than or equal to). The key difference is that while an equation has one specific solution, an inequality typically has infinitely many solutions within a range.

Let's start with a simple example: $2x + 3 > 7$. To solve this, we follow similar steps to solving equations, but with one crucial rule to remember - when we multiply or divide both sides by a negative number, we must flip the inequality sign!

Solving $2x + 3 > 7$:

• Subtract 3 from both sides: $2x > 4$
• Divide both sides by 2: $x > 2$

This means any value of x greater than 2 satisfies our inequality. In real life, this could represent scenarios like "If you need to score more than 7 points total, and you already have 3 points, how many more points do you need?" The answer would be more than 4 points, so if each question is worth 2 points, you need to answer correctly on more than 2 additional questions.

The critical rule about flipping inequality signs when multiplying or dividing by negative numbers often catches students off guard. Consider $-3x < 6$. When we divide both sides by -3, we get $x > -2$ (notice the sign flipped from < to >). This happens because multiplying or dividing by a negative number reverses the order relationship between numbers.

Representing Solutions on Number Lines

Number lines provide a visual representation that makes inequality solutions crystal clear. For $x > 2$, we draw a number line, mark the point 2, and use an open circle (○) to show that 2 is not included in the solution set. Then we shade or draw an arrow extending to the right to show all values greater than 2.

For inequalities with "or equal to" (≤ or ≥), we use a closed circle (●) to indicate that the boundary value is included. For example, $x ≤ -1$ would have a closed circle at -1 with shading extending to the left.

When dealing with compound inequalities like $-3 < x ≤ 5$, we show the solution as a region between two points. This particular inequality means x is greater than -3 (open circle) AND less than or equal to 5 (closed circle). The solution is all values between these two points.

Real-world applications are everywhere! Temperature ranges for storing medicine might be represented as $2°C ≤ T ≤ 8°C$, meaning the temperature must be at least 2°C but no more than 8°C. Speed limits on highways create inequalities too - if the speed limit is 70 mph, then legal speeds satisfy $0 < v ≤ 70$ mph.

Mastering Interval Notation

Interval notation provides a concise mathematical way to express solution sets. It uses brackets and parentheses to indicate whether endpoints are included or excluded. Square brackets [ ] mean the endpoint is included (closed circle on number line), while parentheses ( ) mean the endpoint is excluded (open circle on number line).

For $x > 2$, the interval notation is $(2, ∞)$. The parenthesis at 2 shows it's not included, and infinity (∞) always uses parentheses since infinity isn't a specific number we can reach. For $x ≤ -1$, we write $(-∞, -1]$. The square bracket at -1 indicates it's included in the solution set.

Compound inequalities like $-3 < x ≤ 5$ become $(-3, 5]$ in interval notation. This compact form is especially useful in advanced mathematics and helps avoid confusion when dealing with complex solution sets.

Solving Quadratic Inequalities

Quadratic inequalities involve expressions with $x^2$ terms and require a different approach. The key insight is that quadratic functions create parabolas, and we need to determine where these parabolas are above or below the x-axis.

Let's solve $x^2 - 5x + 6 > 0$. First, we find where the quadratic equals zero by factoring: $x^2 - 5x + 6 = (x-2)(x-3) = 0$. This gives us $x = 2$ and $x = 3$ as our critical points.

These critical points divide the number line into three regions: $x < 2$, $2 < x < 3$, and $x > 3$. We test a value from each region to determine where the original inequality is satisfied:

• For $x < 2$, test $x = 0$: $(0-2)(0-3) = 6 > 0$ ✓
• For $2 < x < 3$, test $x = 2.5$: $(2.5-2)(2.5-3) = -0.25 < 0$ ✗
• For $x > 3$, test $x = 4$: $(4-2)(4-3) = 2 > 0$ ✓

Therefore, $x^2 - 5x + 6 > 0$ when $x < 2$ or $x > 3$, written in interval notation as $(-∞, 2) ∪ (3, ∞)$.

The union symbol ∪ combines separate intervals that are part of the solution set. This represents two distinct regions where our inequality is satisfied.

Advanced Techniques and Applications

When quadratic inequalities don't factor easily, we can use the quadratic formula to find critical points, then apply the same testing method. For inequalities involving fractions with variables in denominators, we must be extra careful about values that make denominators zero, as these create undefined points.

Consider real-world applications: A projectile's height follows a quadratic pattern. If $h(t) = -16t^2 + 64t$ represents height in feet after t seconds, solving $h(t) > 48$ tells us when the projectile is more than 48 feet high. This becomes $-16t^2 + 64t > 48$, or $t^2 - 4t + 3 < 0$ after rearranging. Factoring gives $(t-1)(t-3) < 0$, so the projectile is above 48 feet between 1 and 3 seconds: $1 < t < 3$.

Business applications include profit analysis. If profit $P(x) = -2x^2 + 20x - 18$ where x represents thousands of units sold, solving $P(x) ≥ 32$ determines production levels that achieve at least $32,000 profit.

Conclusion

Inequalities are powerful mathematical tools that help us describe ranges of solutions rather than single answers. We've learned to solve linear inequalities while remembering to flip signs when multiplying or dividing by negatives, represent solutions visually on number lines using open and closed circles, express solutions concisely with interval notation using brackets and parentheses, and tackle quadratic inequalities by finding critical points and testing regions. These skills apply directly to real-world scenarios from physics and engineering to business and economics, making inequalities an essential part of your mathematical toolkit! 🎯

Study Notes

• Linear Inequality Rule: When multiplying or dividing both sides by a negative number, flip the inequality sign

• Number Line Symbols: Open circle (○) for < or >, closed circle (●) for ≤ or ≥

• Interval Notation: Square brackets [ ] include endpoints, parentheses ( ) exclude endpoints

• Infinity Rule: Always use parentheses with ∞ or -∞, never brackets

• Quadratic Inequality Steps: 1) Set equal to zero and solve, 2) Mark critical points, 3) Test regions, 4) Identify solution intervals

• Union Symbol: Use ∪ to combine separate solution intervals

• Key Inequality Symbols: < (less than), > (greater than), ≤ (less than or equal), ≥ (greater than or equal)

• Compound Inequalities: $a < x < b$ means x is between a and b, written as $(a,b)$ in interval notation

• Testing Method: Choose any value in each region created by critical points to determine where inequality is satisfied

• Factoring First: Always try to factor quadratic expressions before applying the quadratic formula for finding critical points