Equations and inequalities

Welcome to Equations and Inequalities!

Hello there! Welcome to one of the most practical chapters in H2 Mathematics. While solving for $x$ might feel like a classic "math class" cliché, this chapter is actually about building the tools to solve real-world problems. Whether it's predicting market trends, balancing chemical equations, or calculating the trajectory of a satellite, it all starts with setting up and solving equations and inequalities.

Don't worry if algebra has felt like a "alphabet soup" in the past. We are going to break this down into clear, logical steps that anyone can follow. Let's dive in!

1. Formulating Equations and Inequalities

Before we can solve a problem, we need to translate "English" into "Math." This is called formulating.

How to approach word problems:

1. Identify the unknowns: Assign letters (like $x, y, z$) to the values you are trying to find.
2. Look for "Relationship Words": Words like "is," "totals," or "the same as" usually mean an equals sign (=). Words like "at most," "exceeds," or "no more than" point toward inequalities ($\le, >, \le$).
3. Write the equations: Organize the information into a system of linear equations if there are multiple unknowns.

Example: A bakery sells 3 cakes and 2 breads for \$120. Another customer buys 1 cake and 5 breads for \$70. Find the price of each.
Let $c$ be the price of a cake and $b$ be the price of bread.
Equation 1: $3c + 2b = 120$
Equation 2: $c + 5b = 70$

Key Takeaway: Always define your variables clearly at the start! It prevents confusion later on.

2. Solving Using the Graphing Calculator (GC)

In H2 Math, the Graphing Calculator (GC) is your best friend. You are expected to know how to use it to solve equations quickly and accurately.

Solving Systems of Linear Equations

For a system like the bakery example above, you can use the PlySmlt2 (Polynomial Root Finder and Simultaneous Equation Solver) app on your TI-84.
1. Select Simultaneous Equation Solver.
2. Input the number of equations and variables.
3. Enter the coefficients and solve.

Solving General Equations Graphically

If you have a complex equation like $e^x = x + 5$:
1. Let $Y_1 = e^x$ and $Y_2 = x + 5$.
2. Graph both functions.
3. Use the CALC -> Intersect function to find where the lines cross. The $x$-coordinate is your solution.

Quick Review: When solving graphically, ensure your "Window" settings are wide enough to see the intersection points!

3. Solving Rational Inequalities

This is where many students trip up. A rational inequality is one where you have a fraction involving $x$, like $\frac{f(x)}{g(x)} > 0$.

The Golden Rule: Never "Cross-Multiply" by $x$!

In equations, you can cross-multiply. In inequalities, you cannot multiply by an expression involving $x$ (like $x-2$) because you don't know if that expression is positive or negative. If it's negative, the inequality sign must flip!

Step-by-Step for $\frac{f(x)}{g(x)} > 0$:

1. Move everything to one side so the other side is zero.
2. Combine into a single fraction with a common denominator.
3. Factorize the numerator and denominator completely.
4. Find the Critical Values: These are the values of $x$ that make the numerator or denominator equal to zero.
5. Use a Sign Table (or the "Wavy Curve" method): Test the regions between your critical values to see if the expression is positive or negative.

Did you know? We exclude values that make the denominator zero because dividing by zero is "undefined"—math's version of a "404 Error"!

Key Takeaway: Always check if your final answer should include the endpoints ($\le$ vs $<$). Denominator critical values are never included.

4. The Modulus Function $|x|$

The modulus (or absolute value) represents the distance of a number from zero. Since distance is always positive, $|x|$ turns negative numbers into positive ones.

Key Modulus Relations to Memorize:

These two rules will save you a lot of time:
1. The "Between" Rule: $|x - a| < b \iff a - b < x < a + b$
Think: The distance from $a$ is small, so we stay close to $a$.
2. The "Outside" Rule: $|x - a| > b \iff x < a - b$ or $x > a + b$
Think: The distance from $a$ is large, so we move far away from $a$ in either direction.

Solving Modulus Equations Algebraicly

If you see $|f(x)| = g(x)$, you can solve it by considering two cases: $f(x) = g(x)$ or $f(x) = -g(x)$.
Warning: Always check your answers back in the original equation! Some solutions might be "extraneous" (fake) because the modulus must result in a non-negative value.

Memory Aid: Less than = And (Inside). Greater than = Or (Outside). Remember "LA-GO"!

5. Solving Inequalities by Graphical Methods

Sometimes, algebra is too messy. This is when we use the Graphical Method. This is specifically useful for modulus inequalities like $|x - 2| > |2x + 1|$.

How to do it:

1. Graph both sides: Let $Y_1 = |x - 2|$ and $Y_2 = |2x + 1|$.
2. Find Intersections: Use your GC to find where the two graphs meet.
3. Interpret the graph: If the question asks for $Y_1 > Y_2$, look for the $x$-values where the graph of $Y_1$ is above the graph of $Y_2$.

Don't worry if this seems tricky at first! Just remember: "Greater than" means "Higher up on the screen."

Key Takeaway: Visualizing the problem often reveals solutions that algebra might hide, especially when dealing with asymptotes or sharp turns in modulus graphs.

Common Mistakes to Avoid

• Sign Flip: Forgetting to flip the inequality sign when multiplying or dividing by a negative number.
• Denominator Dangers: Including the value that makes the denominator zero in your solution set (e.g., writing $x \le 2$ when the fraction is $\frac{1}{x-2}$).
• GC Accuracy: Not providing enough decimal places. Unless otherwise stated, non-exact answers should be to 3 significant figures.
• Squaring: Squaring both sides of an inequality (like $x > -2$) is dangerous because you might create extra incorrect solutions or lose valid ones. Use the sign table method instead!

Summary of Chapter 1.3

To master this chapter, you need to be comfortable with algebraic manipulation (sign tables and modulus rules) and GC techniques (finding intersections). If you can translate a word problem into an equation and then choose the right tool to solve it, you've conquered the hardest part of H2 Functions and Graphs!

* The content provided by thinka is generated by AI and may not always be accurate or up-to-date. Please use it as a supplementary resource and verify with official materials.