Algebra 1 Basics for Beginners
Summary
TLDRThis 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
- đ Solving one-step equations involves isolating the variable by performing the opposite operation on both sides of the equation.
- đą For two-step equations, the order of operations is crucial, and you should eliminate terms step by step, starting with addition or subtraction.
- đ When dealing with equations that include exponents, isolate the term with the exponent and then apply the opposite operation, such as taking the root.
- đ To solve equations with multiple representations of the variable, move all variable terms to one side and constants to the other.
- đ Absolute value equations require setting up two separate equations, one with the expression inside the absolute value being positive and the other negative.
- đ For radical equations, isolate the radical and then square both sides of the equation to eliminate the radical.
- đ Rational equations often involve eliminating fractions by finding a common denominator or cross-multiplying.
- đ Transposing formulas to solve for a specific variable involves moving terms to the other side of the equation using opposite operations.
- đ Solving inequalities follows similar steps to solving equations, but remember to reverse the inequality sign when multiplying or dividing by a negative number.
- đ 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.
- đ 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
𧟠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.
â 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.
đą Solving Multi-Step Equations
This paragraph explains how to solve a multi-step equation (3xÂČ + 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.
âïž 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.
đ 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.
â 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.
𧟠Solving Radical Equations
This paragraph explains solving radical equations (â(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.
â 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.
đ 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.
đ 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.
đ 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.
đ 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.
đŠ 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.
đ¶ 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.
đ 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.
Mindmap
Keywords
đĄOne-step equations
đĄOrder of operations
đĄOpposite operations
đĄMulti-step equations
đĄAbsolute value
đĄRadical equations
đĄRational equations
đĄInequalities
đĄGraphing inequalities
đĄFunctions
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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