Solving Inequalities: Linear, Quadratic, and Wavy Curve Method

1. The Fundamental Rules of Inequalities

Inequalities, like equations, have rules for manipulation. Most are straightforward, but one rule is a frequent source of errors, so it requires special attention.

Let's begin by reviewing these rules.

As you've just read, here are the key takeaways:

1. Addition and Subtraction: You can add or subtract any number on both sides of an inequality without changing the inequality sign.

   • If $a>b$, then $a+c>b+c$ and $a-c>b-c$.

2. Multiplication and Division by a Positive Number: You can multiply or divide both sides by a positive number without changing the inequality sign.

   • If $a>b$ and $c>0$, then $ac>bc$ and $\frac{a}{c}>\frac{b}{c}$.

3. The "Flip Rule": Multiplication and Division by a Negative Number: When you multiply or divide both sides by a negative number, you must reverse the inequality sign.

   • If $a>b$ and $c<0$, then $ac<bc$ and $\frac{a}{c}<\frac{b}{c}$.

4. Variables: You cannot multiply or divide an inequality by a variable unless you know its sign. For example, if you have $\frac{x}{y}>1$, you cannot simply conclude $x>y$, because if $y$ were negative, the sign would need to flip.

Let's practice the most important rule.

2. Solving Linear and Quadratic Inequalities

Solving a linear inequality involves applying the rules above to isolate the variable, just as we did in the exercise.

A quadratic inequality takes the form $a{x}^2+bx+c>0$ (or any other inequality sign). Since these expressions are curves (parabolas), not straight lines, solving them requires finding the points where the expression equals zero and then testing the regions on the number line around those points.

Let's read a clear, step-by-step guide to this process.

The method you just read can be summarized as:

1. Move all terms to one side to get the form $f(x)\le 0$.
2. Solve the corresponding equation $f(x)=0$ to find the critical points.
3. Plot these points on a number line.
4. Test a number from each region to see if the inequality holds true.

While this method is reliable, the "testing" step can be slow. This is where the wavy curve method comes in as a brilliant time-saver.

3. The Wavy Curve Method

The wavy curve method (also known as the method of intervals) is a graphical shortcut to solve polynomial inequalities. It's systematic and visual, which makes it perfect for quickly and accurately solving these problems under exam pressure.

Here's the process:

1. Get to Zero: Ensure the inequality has 0 on one side.
2. Factorize: Factor the polynomial expression.
3. Find Critical Points: Find the values of $x$ that make each factor zero.
4. Plot on Number Line: Mark these critical points on a number line in increasing order.
5. Draw the Curve:
   • Start from the top right. This assumes the expression is written such that the highest power of $x$, when expanded, has a positive coefficient. If not, multiply the whole inequality by -1 and flip the sign.
   • Draw a curve that "crosses" the number line at each critical point.
   • The curve will be above the line in intervals where the expression is positive (+) and below the line where it is negative (-).
6. Select the Solution:
   • If the inequality is $f(x)>0$ or $f(x)\ge 0$, your solution is the intervals where the curve is above the line.
   • If the inequality is $f(x)<0$ or $f(x)\le 0$, your solution is the intervals where the curve is below the line.

Let's see a visual representation of this.

Now, let's read a formal guide on how to apply this method.

Key Points from the Wavy Curve Method:

• Odd Powers: If a factor (x-a) has an odd power (like 1, 3, 5...), the curve crosses the axis at a. This is the standard behavior.
• Even Powers: If a factor (x-a) has an even power (like 2, 4, 6...), the curve touches the axis at a and "bounces back" without crossing. This is because an even power makes the factor always non-negative, so it doesn't change the sign of the overall expression.
• Fractions: For an inequality like $\frac{f(x)}{g(x)}>0$, find the critical points for both $f(x)$ and $g(x)$ and plot them all. Apply the wavy curve method as usual, but remember that the critical points from the denominator $g(x)$ can never be included in the solution (since division by zero is undefined). Use an open circle (o) for these points.

Let's try one more example together.

Problem: Solve $(x-5)(x+1{)}^2(x-2)\le 0$

1. Critical Points: These are -1, 2, and 5.
2. Plot: Mark -1, 2, 5 on the number line.
3. Draw Curve:
   • Start from the top right.
   • At $x=5$ (from factor $(x-5{)}^1$, odd power), the curve crosses the axis.
   • At $x=2$ (from factor $(x-2{)}^1$, odd power), the curve crosses the axis again.
   • At $x=-1$ (from factor $(x+1{)}^2$, even power), the curve touches the axis and bounces back up.
4. Find Solution: We want the intervals where the expression is $\le 0$ (below the line). Looking at the curve, this occurs between 2 and 5. Also, the expression is exactly 0 at the critical points, so we must include them if the inequality allows ($\le$). So, -1, 2, and 5 are part of the solution.
   • The interval is $[2,5]$.
   • The isolated point is $\{-1\}$.
   • Final Solution: $[2,5]\cup \{-1\}$

Conclusion

Today we've built a solid foundation for solving one of the most common question types in algebra. You now have a reliable and fast method to determine the range of values that satisfy a given inequality.

Key Takeaways:

• Always remember to reverse the inequality sign when multiplying or dividing by a negative number.
• Quadratic inequalities can be solved by finding critical points (roots) and testing the intervals between them.
• The wavy curve method is a powerful graphical shortcut for solving polynomial and rational inequalities quickly and accurately. It visualizes where the expression is positive or negative.
• Pay special attention to factors with even powers (where the curve bounces) and critical points from a denominator (which can never be included in the solution).

Preview of the Next Lesson:
We have now concluded our work on linear and quadratic equations and inequalities. In our next lesson, we will tackle a different kind of algebraic problem: simplifying and solving equations involving indices and surds. This will involve mastering the rules of exponents and roots to manipulate complex expressions.