Understanding Linear Inequalities and Exploring Compound Interest

Linear Inequalities and Simple Interest

Ever wondered how to graph areas instead of just lines on a coordinate plane? Linear inequalities use familiar symbols like ≥ (greater than or equal to) and ≤ (less than or equal to) to represent regions on a graph.

When graphing an inequality like y ≥ 2x + 3, first treat it like an equation to find the boundary line. Find the intercepts - in this case (0,3) and $-1.5,0$ - and draw the line. Then shade the appropriate region (above the line for y ≥ 2x + 3).

Simple interest is a straightforward way to calculate growth on investments. The formula is I = PRT, where I is interest earned, P is principal (original amount), R is rate (as a decimal), and T is time in years. The total amount A = P + I.

Quick Tip: When solving simple interest problems, always convert percentages to decimals $6$ and make sure your time is in years (9 months = 0.75 years).

If you know three of the variables in the simple interest formula, you can rearrange it to find the fourth: P = I/RT, R = I/PT, or T = I/PR. This flexibility makes it useful for many financial problems!

Compound Interest and Sequences

Compound interest is what banks actually use - it's interest on your interest! The formula is A = P$1+r/n$^(nt), where n is how many times interest is compounded each year.

Unlike simple interest, compound interest grows exponentially. For example, $10,000 invested at 3.5% for 3 years compounded monthly would grow to much more than with simple interest. The effective interest rate (EIR) tells you the actual percentage gain after compounding.

Sequences help us find patterns in numbers. An arithmetic sequence increases or decreases by a fixed amount each time (like 3, 11, 19, 27...). The general term is a + $n-1$d, where a is the first term and d is the common difference.

Remember This: For reducible interest (like on loans), interest is calculated on the remaining balance after each payment. This means early payments save you more money in the long run!

A geometric sequence multiplies by a constant ratio (like 2, 6, 18, 54...). The formula for the general term is ar^$n-1$, and the sum formula is a$1-r^n$/$1-r$. These patterns appear everywhere from salary increases to population growth!

Sequences and Statistical Measures

The Fibonacci sequence (0, 1, 1, 2, 3, 5, 8...) is a famous pattern where each number equals the sum of the two before it. You'll find this sequence in nature, art, and architecture!

When analyzing data, we often look at univariate data - information about just one variable. For example, test scores from your class would be univariate data. We use several measures to understand this type of data:

• Mean: Add all values and divide by how many there are
• Median: The middle value when data is arranged in order
• Mode: The most frequently occurring value
• Range: The difference between the largest and smallest values

Quartiles divide ordered data into four equal parts. The median (Q2) splits the data in half, while Q1 (lower quartile) and Q3 (upper quartile) divide each half again. These values help us understand how data is distributed.

Pro Tip: When working with a large dataset, calculating quartiles helps you quickly identify unusual values (outliers) that fall far outside the typical range.

Finding these measures gives you a clearer picture of what your data actually means, which is essential for making good decisions based on information.

Bivariate Data and Correlation

Relationships between two variables can tell us powerful stories! Bivariate data examines connections between two different measurements, like height vs. weight or study time vs. test scores.

The relationship between variables can be positive (both increase together), negative (as one increases, the other decreases), or show no relationship at all. The strength of a relationship describes how closely the points follow a pattern.

A scatter plot helps visualize these relationships. Points that form a tight pattern along a line show a strong relationship, while scattered points indicate a weak or nonexistent relationship.

Visual Guide: Think of correlation as measuring how "well-behaved" your data points are. Perfect correlation $r = 1 or -1$ means all points fall exactly on a straight line!

The correlation coefficient $r$ gives a precise measure of relationship strength, ranging from -1 (perfect negative) through 0 (no relationship) to +1 (perfect positive). When analyzing real-world data, finding strong correlations can help you predict outcomes and make better decisions.