Subjects

Algebra 2

•

Dec 7, 2025

•

6 pages

Algebra 2 Chapter 1 Notes: Understanding Equations and Inequalities

Ali

@alisonborowski

Algebra is all about using numbers, variables, and operations to... Show more

1 / 6

value
1.1 Notes /1.3
Algebraic Equations/Expressions.
• Order of operations • Variables
PEMDAS
• expressions VS equations
(#)can't solve Sol

Algebraic Equations and Expressions

Ever wonder what makes equations different from expressions? It's simple - equations have equals signs and can be solved, while expressions just need to be evaluated. Remember to always follow PEMDAS $Parentheses, Exponents, Multiplication/Division, Addition/Subtraction$ when working with either.

When solving equations, follow these key steps:

Clear any fractions or decimals first
Simplify each side (focus on moving variables)
Add or subtract terms to isolate variables
Multiply or divide as needed (avoid dividing by fractions)

For example, solving 53 = 3 $y-2$ - 2 $3y-1$ requires distributing first: 53 = 3y-6-6y+2, then simplifying to 53 = -3y-4. Adding 4 to both sides gives 57 = -3y, which means y = -57/-3 = 19.

Pro Tip: Always check your answers by substituting them back into the original equation to verify you've found the correct solution!

Solving Absolute Value Equations

Absolute value equations might look intimidating, but they're actually straightforward once you understand the technique. Remember that absolute value represents distance from zero, so it's always positive or zero.

The key strategy is to get the absolute value expression by itself first. Then, since |x| = 5 means x = 5 OR x = -5, you'll need to write two separate equations to solve. For example, with |y+3| = 8, you'd write:

y + 3 = 8 $which gives y = 5$
y + 3 = -8 $which gives y = -11$

Be careful with negative signs outside the absolute value. In -2|6-4t| = -16, first isolate the absolute value by dividing both sides by -2, giving you |6-4t| = 8. Then solve both possibilities.

Remember: When your absolute value equals a negative number $like |x| = -5$ , there's no solution because absolute value can never be negative!

Solving Inequalities

Inequalities are like equations with attitude - they tell you when values are greater than, less than, or equal to each other. The most important rule to remember is that when you multiply or divide by a negative number, you must flip the inequality symbol.

To solve an inequality like 4y - 3 < 5y + 2, start by getting variables on one side: 4y - 5y < 2 + 3, which simplifies to -y < 5. Multiply both sides by -1 (and flip the symbol!) to get y > -5.

For fractions in inequalities, multiply both sides by the LCD (least common denominator) to clear the fractions. In the example 3-2x > $x+5$ /3, multiply everything by 3 to get 9-6x > x+5, then solve as normal.

Visual Tip: Always draw your solution on a number line to double-check your work. The direction of your arrow should match your inequality symbol!

Compound and Absolute Value Inequalities

Compound inequalities involve two conditions that must be satisfied at the same time. When you see "AND" conditions like -4 ≤ x < 3, this means x must be in between both values $greater than or equal to -4 AND less than 3$ .

"OR" conditions are different - they're satisfied when either condition is true. For example, x ≥ 5 OR x ≤ -3 means x can be either greater than/equal to 5 OR less than/equal to -3. On a number line, OR graphs typically point in opposite directions.

Absolute value inequalities follow special patterns:

|ax + b| < c becomes -c < ax + b < c $values between -c and c$
|ax + b| ≥ c becomes ax + b ≥ c OR ax + b ≤ -c (values outside the range)

For example, |2x - 1| ≤ 9 means -9 ≤ 2x - 1 ≤ 9, which becomes -4 ≤ x ≤ 5 after solving.

Quick Check: For absolute value inequalities, remember that |x| < c creates an interval, while |x| > c creates two separate regions on the number line.

Relations and Functions

Functions are special relationships where each x-value has exactly one y-value. Think of them as machines that take an input and produce exactly one output. Relations are more general - they're just sets of ordered pairs.

The domain is all possible x-values for your function, while the range contains all possible y-values. Functions can be:

Discrete: Defined only for certain specific values (shown as dots on a graph)
Continuous: Defined for all values within an interval (shown as unbroken lines)

How can you tell if a relation is a function? Use the vertical line test! If any vertical line crosses your graph more than once, it's not a function because that would mean one x-value has multiple y-values.

When evaluating a function like f(x) = x² - 3 at x = -2, simply substitute: f(-2) = (-2)² - 3 = 4 - 3 = 1. This gives you the ordered pair (-2, 1).

Function Check: If you see the same x-value paired with different y-values in a relation, it automatically fails the function test - no graphing required!

Linear Equations and Lines

Linear equations create straight lines on a graph and can be written in different forms. The most common is slope-intercept form $y = mx + b$ , where m represents the slope (rate of change) and b is the y-intercept $where the line crosses the y-axis$ .

To find the slope between two points, use the formula m = $y₂-y₁$ / $x₂-x₁$ . Be careful with vertical lines - their slope is undefined because the denominator would be zero! These are written as x = (some number).

When working with lines, remember:

Parallel lines have the same slope
Perpendicular lines have slopes that are negative reciprocals of each other $like 3 and -1/3$

To write an equation using point-slope form, use y - y₁ = m $x - x₁$ , where (x₁, y₁) is a point on the line and m is the slope. For example, with point (5, -2) and slope 3, you'd write y + 2 = 3 $x - 5$ , which simplifies to y = 3x - 17.

Application Alert: In real-world problems, the slope often represents the rate of change (like cost per hour), while the y-intercept represents the starting value (like an initial fee).

We thought you’d never ask...

What is the Knowunity AI companion?

Our AI companion is specifically built for the needs of students. Based on the millions of content pieces we have on the platform we can provide truly meaningful and relevant answers to students. But its not only about answers, the companion is even more about guiding students through their daily learning challenges, with personalised study plans, quizzes or content pieces in the chat and 100% personalisation based on the students skills and developments.

Where can I download the Knowunity app?

You can download the app in the Google Play Store and in the Apple App Store.

Is Knowunity really free of charge?

That's right! Enjoy free access to study content, connect with fellow students, and get instant help – all at your fingertips.

Can't find what you're looking for? Explore other subjects.

Students love us — and so will you.

4.9/5

App Store

4.8/5

Google Play

The app is very easy to use and well designed. I have found everything I was looking for so far and have been able to learn a lot from the presentations! I will definitely use the app for a class assignment! And of course it also helps a lot as an inspiration.

Stefan S

iOS user

This app is really great. There are so many study notes and help [...]. My problem subject is French, for example, and the app has so many options for help. Thanks to this app, I have improved my French. I would recommend it to anyone.

Samantha Klich

Android user

Wow, I am really amazed. I just tried the app because I've seen it advertised many times and was absolutely stunned. This app is THE HELP you want for school and above all, it offers so many things, such as workouts and fact sheets, which have been VERY helpful to me personally.

Anna

iOS user

I think it’s very much worth it and you’ll end up using it a lot once you get the hang of it and even after looking at others notes you can still ask your Artificial intelligence buddy the question and ask to simplify it if you still don’t get it!!! In the end I think it’s worth it 😊👍 ⚠️Also DID I MENTION ITS FREEE YOU DON’T HAVE TO PAY FOR ANYTHING AND STILL GET YOUR GRADES IN PERFECTLY❗️❗️⚠️

Thomas R

iOS user

Knowunity is the BEST app I’ve used in a minute. This is not an ai review or anything this is genuinely coming from a 7th grade student (I know 2011 im young) but dude this app is a 10/10 i have maintained a 3.8 gpa and have plenty of time for gaming. I love it and my mom is just happy I got good grades

Brad T

Android user

Not only did it help me find the answer but it also showed me alternative ways to solve it. I was horrible in math and science but now I have an a in both subjects. Thanks for the help🤍🤍

David K

iOS user

The app's just great! All I have to do is enter the topic in the search bar and I get the response real fast. I don't have to watch 10 YouTube videos to understand something, so I'm saving my time. Highly recommended!

Sudenaz Ocak

Android user

In school I was really bad at maths but thanks to the app, I am doing better now. I am so grateful that you made the app.

Greenlight Bonnie

Android user

I found this app a couple years ago and it has only gotten better since then. I really love it because it can help with written questions and photo questions. Also, it can find study guides that other people have made as well as flashcard sets and practice tests. The free version is also amazing for students who might not be able to afford it. Would 100% recommend

Aubrey

iOS user

Best app if you're in Highschool or Junior high. I have been using this app for 2 school years and it's the best, it's good if you don't have anyone to help you with school work.😋🩷🎀

Marco B

iOS user

THE QUIZES AND FLASHCARDS ARE SO USEFUL AND I LOVE THE SCHOOLGPT. IT ALSO IS LITREALLY LIKE CHATGPT BUT SMARTER!! HELPED ME WITH MY MASCARA PROBLEMS TOO!! AS WELL AS MY REAL SUBJECTS ! DUHHH 😍😁😲🤑💗✨🎀😮

Elisha

iOS user

This app is phenomenal down to the correct info and the various topics you can study! I greatly recommend it for people who struggle with procrastination and those who need homework help. It has been perfectly accurate for world 1 history as far as I’ve seen! Geometry too!

Paul T

iOS user

Stefan S

iOS user

Samantha Klich

Android user

Anna

iOS user

Thomas R

iOS user

Brad T

Android user

Not only did it help me find the answer but it also showed me alternative ways to solve it. I was horrible in math and science but now I have an a in both subjects. Thanks for the help🤍🤍

David K

iOS user

Sudenaz Ocak

Android user

In school I was really bad at maths but thanks to the app, I am doing better now. I am so grateful that you made the app.

Greenlight Bonnie

Android user

Aubrey

iOS user

Best app if you're in Highschool or Junior high. I have been using this app for 2 school years and it's the best, it's good if you don't have anyone to help you with school work.😋🩷🎀

Marco B

iOS user

Elisha

iOS user

Paul T

iOS user

Subjects

Algebra 2

•

Dec 7, 2025

•

6 pages

Algebra 2 Chapter 1 Notes: Understanding Equations and Inequalities

Ali

@alisonborowski

Algebra is all about using numbers, variables, and operations to solve problems. These notes cover essential concepts from algebraic equations to functions, giving you the tools to solve a variety of mathematical challenges.

Sign up to see the contentIt's free!

Access to all documents

Improve your grades

Join milions of students

By signing up you accept Terms of Service and Privacy Policy

Algebraic Equations and Expressions

When solving equations, follow these key steps:

Clear any fractions or decimals first
Simplify each side (focus on moving variables)
Add or subtract terms to isolate variables
Multiply or divide as needed (avoid dividing by fractions)

For example, solving 53 = 3 $y-2$ - 2 $3y-1$ requires distributing first: 53 = 3y-6-6y+2, then simplifying to 53 = -3y-4. Adding 4 to both sides gives 57 = -3y, which means y = -57/-3 = 19.

Pro Tip: Always check your answers by substituting them back into the original equation to verify you've found the correct solution!

Sign up to see the contentIt's free!

Access to all documents

Improve your grades

Join milions of students

By signing up you accept Terms of Service and Privacy Policy

Solving Absolute Value Equations

y + 3 = 8 $which gives y = 5$
y + 3 = -8 $which gives y = -11$

Be careful with negative signs outside the absolute value. In -2|6-4t| = -16, first isolate the absolute value by dividing both sides by -2, giving you |6-4t| = 8. Then solve both possibilities.

Remember: When your absolute value equals a negative number $like |x| = -5$ , there's no solution because absolute value can never be negative!

Sign up to see the contentIt's free!

Access to all documents

Improve your grades

Join milions of students

By signing up you accept Terms of Service and Privacy Policy

Solving Inequalities

Visual Tip: Always draw your solution on a number line to double-check your work. The direction of your arrow should match your inequality symbol!

Sign up to see the contentIt's free!

Access to all documents

Improve your grades

Join milions of students

By signing up you accept Terms of Service and Privacy Policy

Compound and Absolute Value Inequalities

Absolute value inequalities follow special patterns:

|ax + b| < c becomes -c < ax + b < c $values between -c and c$
|ax + b| ≥ c becomes ax + b ≥ c OR ax + b ≤ -c (values outside the range)

For example, |2x - 1| ≤ 9 means -9 ≤ 2x - 1 ≤ 9, which becomes -4 ≤ x ≤ 5 after solving.

Quick Check: For absolute value inequalities, remember that |x| < c creates an interval, while |x| > c creates two separate regions on the number line.

Sign up to see the contentIt's free!

Access to all documents

Improve your grades

Join milions of students

By signing up you accept Terms of Service and Privacy Policy

Relations and Functions

The domain is all possible x-values for your function, while the range contains all possible y-values. Functions can be:

Discrete: Defined only for certain specific values (shown as dots on a graph)
Continuous: Defined for all values within an interval (shown as unbroken lines)

When evaluating a function like f(x) = x² - 3 at x = -2, simply substitute: f(-2) = (-2)² - 3 = 4 - 3 = 1. This gives you the ordered pair (-2, 1).

Function Check: If you see the same x-value paired with different y-values in a relation, it automatically fails the function test - no graphing required!

Sign up to see the contentIt's free!

Access to all documents

Improve your grades

Join milions of students

By signing up you accept Terms of Service and Privacy Policy

Linear Equations and Lines

When working with lines, remember:

Parallel lines have the same slope
Perpendicular lines have slopes that are negative reciprocals of each other $like 3 and -1/3$

Application Alert: In real-world problems, the slope often represents the rate of change (like cost per hour), while the y-intercept represents the starting value (like an initial fee).