Algebra 1 Basics for Beginners

Summary

This Ultimate Algebra video offers a comprehensive guide to solving Algebra 1 problems efficiently. It covers solving one-step equations, multi-step equations, inequalities, and graphing. The video emphasizes mastering basic operations like addition, subtraction, multiplication, and division to isolate variables. It also teaches how to handle absolute values and radicals in equations. Practical examples include determining the number of gallons per box and Michael's age from word problems. The video concludes with a discussion on functions, explaining what qualifies as a function and providing examples. Viewers are encouraged to deepen their understanding with Ultimate Algebra's full course.

Takeaways

• 📘 Solving one-step equations involves isolating the variable by performing the opposite operation on both sides of the equation.
• 🔢 For two-step equations, the order of operations is crucial, and you should eliminate terms step by step, starting with addition or subtraction.
• 📐 When dealing with equations that include exponents, isolate the term with the exponent and then apply the opposite operation, such as taking the root.
• 🔄 To solve equations with multiple representations of the variable, move all variable terms to one side and constants to the other.
• 📊 Absolute value equations require setting up two separate equations, one with the expression inside the absolute value being positive and the other negative.
• 🌐 For radical equations, isolate the radical and then square both sides of the equation to eliminate the radical.
• 📑 Rational equations often involve eliminating fractions by finding a common denominator or cross-multiplying.
• 📈 Transposing formulas to solve for a specific variable involves moving terms to the other side of the equation using opposite operations.
• 📉 Solving inequalities follows similar steps to solving equations, but remember to reverse the inequality sign when multiplying or dividing by a negative number.
• 📋 Graphing inequalities on a number line involves placing an open or closed circle at the critical point and drawing an arrow in the direction of the inequality sign.
• 📝 Word problems often require identifying key values and setting up equations to find the unknown, such as calculating the number of gallons per box.

Q & A

How do you solve a one-step equation like x + 2 = 5?

To solve a one-step equation, isolate the variable by performing the opposite operation on both sides. For x + 2 = 5, subtract 2 from both sides to get x = 3.

What is the order of operations and how does it help in solving equations?

The order of operations is a rule in mathematics that dictates which operations to perform first. It helps in solving equations by determining the sequence of operations to reverse when isolating the variable.

How do you solve a two-step equation like 2x + 3 = 11?

First, eliminate any addition or subtraction terms by performing the opposite operation. Then, isolate the variable by dividing both sides by the coefficient of the variable.

What is the process for solving multi-step equations?

Solve multi-step equations by isolating the variable on one side of the equation. This involves reversing the order of operations, starting with addition or subtraction, then multiplication or division, and finally dealing with exponents.

How do you handle equations with the variable represented more than once, like 4x + 5 = 9 + 2x?

Move all variable terms to one side and constants to the other side. For 4x + 5 = 9 + 2x, subtract 2x from both sides to combine like terms, then solve for x.

How do you solve absolute value equations?

Treat the absolute value as both positive and negative of the value on the other side of the equation. Solve both resulting equations to find the possible values of the variable.

What is the key to solving rational equations?

To solve rational equations, eliminate the fractions by finding a common denominator or by cross-multiplying, then solve for the variable.

How do you approach solving inequalities?

Solve inequalities similarly to equations, but remember to reverse the inequality sign when multiplying or dividing by a negative number.

Can you explain how to graph an inequality on a number line?

Graph an inequality by placing a circle at the critical point and drawing an arrow in the direction indicated by the inequality sign. Use an open circle for 'greater than' or 'less than' and a closed circle for 'greater than or equal to' or 'less than or equal to'.

How do you solve word problems that involve packaging items into groups, like shipping 2,500 gallons into boxes?

Identify the total amount, the number of groups (boxes), and any remainder. Set up an equation with the group size as the variable, and solve for the variable to find the amount per group.

What is the definition of a function in algebra?

A function in algebra is a relation where each input value corresponds to exactly one output value. No input value can have multiple output values.

Outlines

00:00

🧮 Introduction to Solving Algebra 1 Problems

This paragraph introduces the video, explaining that it focuses on solving Algebra 1 questions in the simplest way. Viewers are encouraged to explore the Ultimate Algebra course for a more in-depth understanding. The first problem, a simple one-step equation (x + 2 = 5), is solved by isolating x through subtraction of 2 from both sides, resulting in x = 3.

05:01

➗ Solving Two-Step Equations

The second paragraph demonstrates solving a two-step equation (2x + 3 = 11). The process is explained step by step using the reverse order of operations, beginning by eliminating addition (subtracting 3) and then removing multiplication (dividing by 2), resulting in x = 4.

10:04

🔢 Solving Multi-Step Equations

This paragraph explains how to solve a multi-step equation (3x² + 8 = 20). The order of operations is reversed, first subtracting 8, then dividing by 3, and finally taking the square root to solve for x, giving x = 2.

15:06

✖️ Solving Equations with X on Both Sides

The fourth paragraph introduces equations where x appears on both sides (4x + 5 = 9 + 2x). The approach is to move all terms containing x to one side, and then solve the resulting two-step equation. The solution yields x = 2.

20:10

📐 Solving Absolute Value Equations

This paragraph covers solving absolute value equations. In the example (|x + 3| = 7), the equation is split into two cases: x + 3 = 7 and x + 3 = -7. Solving both cases results in x = 4 and x = -10.

➕ Handling Absolute Value with Additional Terms

Here, a more complex absolute value equation is presented (|x + 1| + 6 = 9). The first step is to isolate the absolute value by subtracting 6 from both sides. Then, the equation is split into two cases, resulting in x = 2 and x = -4.

🧮 Solving Radical Equations

This paragraph explains solving radical equations (√(x + 3) = 3). First, the equation is simplified by isolating the radical, squaring both sides, and then solving the resulting one-step equation to find x = 6.

➗ Solving Rational Equations

The focus here is on solving rational equations (4/(x - 5) = 3/x). Cross multiplication is used to eliminate the fractions, resulting in a linear equation, which is solved to find x = -5.

📏 Changing the Subject of a Formula

This paragraph explains how to solve for x in a formula (y = mx + b). Using the reverse order of operations, the first step is to move b by subtraction, and then divide by m, resulting in x = (y - b)/m.

📉 Solving Inequalities

This section covers solving inequalities (-3x + 1 > 7). The process is similar to solving equations, but it includes the rule that when dividing by a negative number, the inequality sign flips. The solution is x < -2.

🔀 Solving Compound Inequalities

Here, the method of solving compound inequalities (-3 < x + 8 < 20) is explained. The same operation (subtracting 8) is applied to all three parts of the inequality, leading to the solution -11 < x < 12.

📊 Graphing Inequalities on a Number Line

This paragraph introduces graphing inequalities, using the example x > -4. The number line is drawn with an open circle at -4 (indicating x is greater but not equal) and an arrow pointing to the right.

📦 Solving Word Problems with Two-Step Equations

This section walks through a two-step word problem involving the packaging of 2,500 gallons of product into 20 boxes, with 100 gallons left over. By setting up and solving the equation 20x + 100 = 2,500, the solution for x (gallons per box) is 120.

👶 Solving Age-Related Word Problems

This paragraph explains how to solve an age-related word problem: 'Five added to thrice Michael's age is 50.' The equation 5 + 3x = 50 is solved step by step, resulting in Michael's age being 15 years.

🔁 Identifying Functions from Relations

The final paragraph explains how to identify if a relation is a function. It emphasizes that each input value must have only one output value, and gives an example where one input has two outputs (making it not a function). The answer to the example question is C.

Keywords

💡One-step equations

One-step equations involve solving for the unknown variable in just one step, such as using addition, subtraction, multiplication, or division. In the video, these are foundational for understanding algebra, with examples like 'x + 2 = 5,' where subtracting 2 from both sides quickly solves for x.

💡Order of operations

The order of operations is the sequence in which different mathematical operations are performed. This is crucial in solving multi-step algebra problems. The video highlights its importance, demonstrating how addition and multiplication must be approached in reverse when solving equations like '2x + 3 = 11.'

💡Opposite operations

Opposite operations refer to the idea of using inverse functions (like addition vs. subtraction or multiplication vs. division) to move terms from one side of an equation to the other. In the video, it is emphasized as a fundamental technique for simplifying equations, such as subtracting 3 from both sides of 'x + 3 = 7.'

💡Multi-step equations

Multi-step equations require more than one operation to isolate the variable. The video uses '3x^2 + 8 = 20' as an example, explaining how to break down the steps by reversing the order of operations, starting with subtraction, then division, and finally addressing the exponent.

💡Absolute value

Absolute value refers to the non-negative value of a number, regardless of its sign. The video shows how to solve absolute value equations by considering both the positive and negative solutions, as in 'x + 3 = 7 and x + 3 = -7.'

💡Radical equations

Radical equations involve a variable inside a radical (such as a square root). The video demonstrates how to handle these, explaining the process of isolating the radical before squaring both sides to eliminate the square root, as shown in '√(x + 3) = 3.'

💡Rational equations

Rational equations feature variables in the denominator of fractions. The video explains how to solve these by cross-multiplying, as in the example '4/(x - 5) = 3/x,' showing how to eliminate fractions and simplify.

💡Inequalities

Inequalities are mathematical expressions that use symbols like '>' or '<' to compare values. In the video, inequalities are solved similarly to equations, with special attention to how multiplying or dividing by a negative number flips the inequality sign, as seen in '-3x + 1 > 7.'

💡Graphing inequalities

Graphing inequalities involves representing the solutions of an inequality on a number line. The video illustrates this concept by explaining how to use open or shaded circles depending on whether the inequality includes equality (e.g., 'x > -4' is represented with an open circle).

💡Functions

A function is a relation where each input has exactly one output. The video discusses how a relation can be checked for being a function, showing that if an input like 'x = 3' has two outputs (e.g., 6 and 8), it is not a function, reinforcing the concept that functions have unique outputs for each input.

Highlights

Introduction to solving Algebra 1 questions using the easiest methods.

Explanation of solving one-step equations by isolating the variable.

Demonstration of how to perform the opposite operation to isolate variables.

Emphasis on the importance of mastering one-step equations for math tests.

Guide on solving two-step equations, such as 2x + 3 = 11.

Introduction to the order of operations and its role in solving equations.

Process of solving multi-step equations like 3x^2 + 8 = 20.

Explanation of how to handle equations with variables represented twice.

Approach to solving absolute value equations.

Method for solving radical equations with variables inside a radical sign.

Technique for solving rational equations with variables in the denominator.

How to change the subject or transpose formulas to solve for a specific variable.

Process of solving inequalities, including handling negative numbers.

Guide on solving combined inequalities.

Instructions on graphing inequalities on a number line.

Example of solving a word problem involving packaging gallons into boxes.

Explanation of how to quickly solve word problems using a two-step equation.

Method for solving age-related word problems using algebraic equations.

Criteria for determining whether a relation is a function or not.

Identification of relations that are not functions due to multiple output values for the same input.

Encouragement to get the full course for mastering algebra.

Transcripts

welcome to another video from Ultimate

algebra.com in this video we will be

looking at how to answer Algebra 1

questions the easiest way please this

video will not be exhaustive for a

complete course deeper dive with more

examples please check out our ultimate

algebra course at ultimate algebra.com

let's Dive Right

In question 1 x + 2 = 5 solve for x the

first thing we are looking at is how to

solve one-step equations you have to be

good at solving one-step equations in

order to pass any math test the idea of

solving equations is to move everything

with the X to the other side of the

equation by performing the opposite

operation on it we want only the X to be

on one side of the equation so here we

want to move the plus two to the other

side of the equation we do that by

performing the opposite operation on

both sides of the equation the opposite

operation of addition is subtraction the

opposite operation of multiplication is

division the opposite operation of

exponent is the root or radical we know

the opposite operation of addition is

subtraction so we will subtract two from

both sides once the opposite operation

is done those two numbers simply cancels

out so here the two will cancel out 5 -

2 is 3 so X will be equal to 3 we have a

free complete video on solving equations

with a lot more

examples please check it out with the

link in the description for more let's

move on to question

two question 2 2x + 3 = 11 solve for x

here we are looking at solving 2ep

equations we said earlier that the whole

idea of solving equations is to get get

rid of everything and leave the X on one

side of the equation for this question

we will see that we have to get rid of

the multiplication by 2 in the plus

three 2x is the same as 2 * X when there

are more than one operations we use the

idea of the reversal of the order of

operations to know which one to get rid

of first so here is the order of

operations if you are not familiar with

the order of operations please revise

our videos on it it's very important

knowing the order of operations will

make solving of equations super easy we

can see that in the reversal of the

order of operations that's from bottom

to top we have addition before

multiplication so we will get rid of the

plus three first we get rid of the plus

three by performing the opposite

operation on it so we will subtract

three from both sides the three will

cancel out 11 - 3 is 8 so we have 2 x x

= 8 next we will get rid of the

multiplication by two by dividing both

sides by two since division is the

opposite of

multiplication the two will cancel out 8

/ 2 is 4 therefore x =

4 question three 3x^2 + 8 = 20 solve for

x we are looking at solving multi-step

equations the process of solving is just

like solving twostep equations we want

to get the X on one side of the equation

in order to do that we have to get rid

of the multiplication by three the

exponent 2 and the Plus 8 we will use

the reversal of the order of operation

to know which one to perform first let's

bring our order of operations so here is

our order of

operations in the reversal we will

notice that we have to to do the Plus 8

first then we will do the multiplication

by three and then we do the exponent two

we will get rid of the Plus 8 by

performing the opposite operation on it

subtract eight from both sides the eight

will cancel out 20 - 8 is 12 so now we

have 3x^2 = 12 next we have to get rid

of the multiplication by three we do the

opposite operation we divide both sides

by three

the 3 will cancel out 12 / 3 is 4 so now

we have x^2 = 4 we finally have to get

rid of the exponent two we do the

opposite operation the opposite

operation of squared is square root we

find the square root of both sides this

will cancel out the square root of 4 is

2 therefore x =

2

question 4 4x + 5 = 9 + 2x solve for x

in our previous questions we have been

having the X represented only once

example 4x + 5 equals 9 but here we have

the X represented twice in a case like

this we want to move the X values to one

side of the equation and work on it so

here you can choose to move the 4X or 2X

I'll move the 2x to the other side of

the equation to do that since the 2x is

adding you will subtract 2x from both

sides of the equation the 2x will cancel

out now 4x - 2x is 2x so we have 2X + 5

= 9 we now have a familiar two-step

equation which you should be able to

solve if you've watched the previous

questions but let's go over it we want

to get rid of all the numbers at

attached to the X so we can have the X

alone on one side of the equation to

achieve this we know that we must get

rid of the time 2 in the + 5 let's bring

our order of operations please you don't

need to be writing the order of

operations in your Solutions we are

using it for teaching purpose we are

using the reversal of the order of

operations so we are working from the

bottom up we will see that in this form

we must do the addition first before the

multiplication

to get rid of the plus 5 we must perform

the opposite operation on both sides of

the equations so we will subtract five

from both sides the five cancels out 9 -

5 is 4 we now have 2x = 4 next we will

get rid of the multiplication by two we

do this by performing the opposite

operation on both sides of the equation

so we will divide both sides by two the

two will cancel out 4 / 2 is 2 therefore

x =

2 question five the absolute value of x

+ 3 = 7 find X for absolute value

equations we equate the absolute value

to the positive and negative of what is

on the other side of the equation here

we will equate the absolute value to

positive and negative of the 7 so we

have x + 3 = 7 and x + 3 = -7 we solve

both equations for the first one we

subtract three from both sides the three

will cancel out 7 - 3 is 4 for the

second one we subtract three from both

sides the three will cancel out -7 - 3

is -10 so our answer is x = 7 and and X

=

-10 question 6 the absolute value of x +

1 + 6 = 9 find X in the previous

question everything on the left side was

in the absolute value Marks here the

trick is that the plus 6 is not in the

absolute value you need to remove

everything that is not in the absolute

value Mark and get only the absolute

value on one side for you can equate to

the negative and positive so we will

first start by subtracting six from both

sides the six will cancel out 9 - 6 is 3

now we have the absolute value of x + 1

= 3 we have only the absolute value on

one side of the equation so we can

equate x + 1 to the positive and

negative of what is here as usual we

have x + 1 = -3 and x + 1 = POS 3 we

subtract one from all sides the one will

cancel out for this one -3 - 1 will be

-4 so x = -4 for this one 3 - 1 will be

2 so x = 2 so x = -4 or x =

2 question 7 s < TK x + 3 - 2 = 1 find X

we are looking at radical

equations a radical equation is an

equation in which the variable is

contained inside a radical or root sign

here we see that the x is under the

square root sign similar to what we did

for absolute value equations we want the

isolate the radical on one side of the

equations we add two to both sides the

two will cancel out 1 + 2 will give us

three now we have the square < TK of x +

3 = 3 next we want to eliminate the

square root by squaring both sides of

the equation for the left side the

square root will cancel out the square

we just get x + 3 3^ 2 is 9 we have a

one-step equation we subtract three from

both sides the three will cancel out 9 -

3 is 6 therefore x =

6

question 8 4 / the quantity x - 5 = 3 /

X find X we are looking at rational

equations for rational equations we have

the x or variable in the

denominator the first step is to remove

the fractions typically we will use the

least common denominator method but for

this question we can just cross multiply

4 * X is 4 x 3 * x - 5 is 3 x - 15 we

expanded it 3 * X is 3x and 3 * -5 is

-15 next we want to isolate the X on one

side of the equation we subtract 3x from

both sides the 3x will cancel out 4x -

3x is 1 X or simply X so we have X = -5

as our answer

question nine solve for x given that yal

mx + b here we are looking at change of

subject or transposing of

formula this is the slope intercept form

of the equation of a line when you are

given any formula you should be able to

find any of the

variables this is nothing different from

what we've been doing so far to solve

for x we we want to have the X on one

side of the equation and everything else

on the other side of the equation to do

that we have to move the times M and the

plus b we will use the reversal of the

order of operation to know which one to

perform first let's bring our order of

operations in the reversal we will

notice that we have to do the plus b

first then we will do the multiplication

by m we will get rid of the plus b by

performing the opposite operation on it

subtract B from both sides the B will

cancel out you cannot subtract y minus B

because they are dissimilar as we

learned in addition and subtractions in

algebra so we will have y - B = MX next

we have to get rid of the multiplication

by m we do the opposite operation we

divide both sides by m the m will cancel

out there's nothing we can reduce on

this other side so x = y - B all over

M question 10 -3x + 1 is greater than 7

solve for x here we are looking at

solving inequalities the process is the

same as solving

equations there is a slight difference

when you multiply or divide by a

negative we want to get rid of

everything and leave it X on one side of

the inequality sign for this question we

will see that we have to get rid of the

multiplication by -3 and the plus one

let's bring our order of

operations we can see that in the

reversal of the order of operations

that's from bottom to top we have

addition before

multiplication so we will get rid of the

plus first we get rid of the plus one by

performing the opposite operation on it

so we will subtract one from both sides

the one will cancel out 7 - 1 is 6 so we

have -3x is greater than 6 next we have

to divide both sides by the -3 so we can

get the X by itself in inequalities when

you divide or multiply by negative the

inequality changes so here the greater

than becomes less than please take note

of this nearly all wrong answers are

because of this mistake

now the -3 will cancel out 6 / -3 is -2

therefore X is less than

-2 question 11 solve the inequality -3

less than x + 8 less than 20 here we are

looking at combined

inequalities this question is the same

as -3 less than x + 8 and x + 8 less

than 20 we just com combine them the

solution is exactly the same instead of

having two sides you now have three

sides to get the X by itself we have to

subtract eight from all three sides the8

will cancel out here -3 - 8 is

-11 then 20 - 8 is 12 so our answer is

-1 less than x less than 12 question 12

graph the inequality X greater than

-4 here we are looking at graphing

inequalities let's bring our number line

when graphing inequalities the first

thing is to locate your point which will

be the number here it is -4 then you'll

draw a shaded or unshaded circle at the

point if you have less than or greater

than then the circle will not be shaded

if you have less than or equal to or

greater than or equal to you will use

the Shaded Circle

so basically if it has an equal to you

will shade in this case since it's just

greater than we will not shade the

circle then finally we draw an arrow the

easiest way to get this always right

without thinking is to make sure your

inequality is in the form variable

inequality sign and then the number in

that form the direction of your arrow

will be the same as the direction of the

inequality sign here since this is in

that form we will draw our Arrow facing

here and we are

done question 13 in order to ship 2,500

gallons of a product to another country

Stan shipping company had to package the

gallons into boxes if they had 20

package boxes and 100 gallons left that

are not in boxes how many gallons were

in a box this kind of two-step equation

word problem is is very common we are

first going to solve it in details for

teaching purpose then I'll show you how

you can solve it in less than 10 seconds

on an actual test you have three values

in these type of questions let's write

them down we have 2,500 gallons we have

20 boxes and finally we have 100 gallons

the 2,500 gallons represents the total

so we have equal

2,500 the 20 box is what I call the

group The gallons have been grouped into

boxes for most questions the group can

also be identified as the number that

represents something different from the

other two numbers so here 2,500

represents gallons the 100 also

represent gallons but the 20 represent

boxes so the 20 will be the group the

group is the one with the X so we will

have

20x we can now add the 100 gallons left

to to the equation and solve for the X

in this two-step equation subtract 100

from both sides these will cancel out

2500 - 100 will be

2,400 we now have 20x =

2,400 divide both sides by 20 the 20

will cancel out 2,400 ided 20 will be

120 this means there were 120 gallons in

each box the hard part of this question

is to be able to pull out the values

from the word

problem we went through a detailed

solution for teaching purpose let's look

at how you can speed up solving this

question first there's absolutely no

reason to write this part if you know

what you're doing you can go straight to

writing your two-step equations we have

our group 20x our other

+ and our total

2,500 then you can solve the two-step

equation

equation an even faster method to solve

this question in less than 10 seconds is

to do 2,500 minus 100 divided by 20 on

your calculator to get

120 we did the total minus other divided

by the group a big caution when using

fast methods is to note that they are

very specific to specific questions and

little twists to the question can let

you get it wrong so when in doubt use

the longer

methods

question 14 five added to Thrice

Michael's age is 50 how old is Michael

for question like this it is good

practice to start with identifying your

unknown value and representing it by a

letter let's say x the unknown value is

usually what the question is asking you

to find here it is Michael's age let's

represent it with X now we just

translate we know added to is addition

Thrice is three times so Thrice

Michael's age is 3x and is means equal

to so this is 5 + 3x = 50 you can now

solve the two-step equation we subtract

five from both sides the five will

cancel out 15 - 5 will be

45 so we have 3x =

45 next we will divide both sides by

three this will cancel out 4 5 / 3 is 15

therefore x = 15 Michael's age is 15

years question 15 which of the relations

below is not a function to answer this

question let's go through a few things

about functions when you have a notation

example 25 the two represents the input

value that's x value the five represents

the output value that's the Y value this

is the same as what we will learn in

graphing points for a relation to be a

function no input value can have

multiple output values also all input

values must have output values let's

look at examples here the input values

are these and the output values are

these note that each input value

corresponds to only one output value

although three is an output value

without any input value corresponding to

it it is still a function this could

have been written in this form we wrote

each of the input values and what output

value they correspond to now let's look

at this relations this relation is not a

function because the input two has more

than one output value that's seven and

five no input value can have more than

one output value this could have been

written this way notice we have two

written twice with different y values

one is seven and the other is five

finally let's look at this

relations this is a function multiple

input values can go to the same output

value so here although the input value

four and the input value five both gives

eight it is a function again this could

have been written this way notice we

have our y value 8 written twice with

this information let's answer answer our

question the answer is c c is not a

function the input value three has two

output values that's six and8 for a

function no input value can have more

than one output

values Happy Thanksgiving get the full

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