4 Steps to Solving Multi-Step Inequalities | 7.EE.B.4 💚

Summary

In 'The Magic of Math' video, the host teaches how to solve multi-step inequalities using four key steps: clear parentheses, isolate the variable, get a coefficient of one, and graph the solution. The tutorial employs prior knowledge of equations to guide viewers through examples, emphasizing the importance of maintaining inequality properties throughout the process. The lesson aims to make mastering math approachable and engaging.

Takeaways

• 📘 The lesson focuses on solving multi-step inequalities, which requires understanding and applying prior knowledge of solving equations and inequalities.
• 🔍 There are four key steps to solving multi-step inequalities: clearing parentheses, isolating the variable term, simplifying the variable to a coefficient of one, and graphing the solution.
• 👉 Step one involves using the distributive property to clear parentheses if necessary.
• 🔄 Step two circles the variable term and adds or subtracts values from both sides to isolate the variable.
• 🔄 Step three circles the variable and involves multiplying or dividing by a value to simplify the variable's coefficient to one.
• 📊 Step four is graphing the solution on a number line, using appropriate symbols for the inequality type (e.g., open or closed circles).
• ⚠️ It's important to remember that operations in front of terms can indicate whether they are positive or negative, affecting the direction of the inequality.
• ➗ When dividing both sides of an inequality by a negative number, the inequality symbol must be reversed.
• 📐 The process includes creating 'zero pairs' to isolate the variable term, which involves performing the inverse operation to what is already present.
• 📈 The video provides examples of solving inequalities with different operations, emphasizing the need to maintain the properties of equality when manipulating both sides of an inequality.
• 📚 The lesson encourages viewers to practice solving inequalities by pausing the video and attempting the problems themselves, then checking their work against the provided solutions.

Q & A

What is the main topic of the 'Magic of Math' video?

The main topic of the video is solving multi-step inequalities.

What are the four steps to solving multi-step inequalities as outlined in the video?

The four steps are: 1) Clear parentheses by performing the distributive property if necessary, 2) Circle the variable term and add or subtract a value from both sides to isolate it, 3) Circle the variable and multiply or divide a value from both sides to get a coefficient of one, and 4) Graph the solution.

Why might steps one and two not be necessary when solving multi-step inequalities?

Steps one and two might not be necessary if the inequality does not have parentheses that require clearing or if the variable term is already isolated.

What is the end goal when solving for the variable in an inequality?

The end goal is to have the variable with a coefficient of one, so you can determine what the variable is less than, greater than, or equal to.

How does the video suggest handling negative values when solving inequalities?

When multiplying or dividing by a negative value, you must reverse the inequality symbol to maintain the correct relationship.

What symbol should be used on a number line to represent a value that the variable can be equal to?

A closed circle should be used on the number line to represent a value that the variable can be equal to.

What symbol should be used on a number line to represent a value that the variable cannot be equal to?

An open circle should be used on the number line to represent a value that the variable cannot be equal to.

How does the video suggest identifying the variable term in an inequality?

The video suggests identifying the variable term by looking at what operation is applied to it and noting whether it's positive or negative.

What is the purpose of creating a 'zero pair' when solving inequalities?

Creating a 'zero pair' is done to isolate the variable term by eliminating the constant that is added or subtracted from it.

What should you do when you encounter parentheses in an inequality that requires solving?

When encountering parentheses, you should perform the distributive property to clear them before proceeding with the other steps.

How does the video demonstrate the process of solving a multi-step inequality?

The video demonstrates the process by walking through several examples, showing each step from isolating the variable to graphing the solution on a number line.

Outlines

00:00

📚 Introduction to Solving Multi-Step Inequalities

The script begins with an introduction to a lesson on solving multi-step inequalities, emphasizing the connection between solving equations and inequalities. The instructor outlines a four-step process to tackle these problems: clearing parentheses, isolating the variable term, simplifying the variable to have a coefficient of one, and graphing the solution. The lesson encourages students to apply prior knowledge and introduces the concept of zero pairs and the importance of maintaining the properties of equality when manipulating inequalities.

05:00

🔍 Detailed Steps and Examples for Solving Inequalities

This paragraph delves into the specifics of solving multi-step inequalities with detailed examples. It covers the process of eliminating parentheses, if present, and then isolating the variable term by creating zero pairs. The instructor demonstrates how to reverse operations to simplify the inequality, such as dividing by a negative number, which requires flipping the inequality sign. Each example is followed by a step to graph the solution set on a number line, using open or closed circles to indicate inclusivity or exclusivity of the endpoint, and shading the appropriate region to represent the solution set.

10:02

📈 Final Steps and Conclusion of the Lesson

The final paragraph wraps up the lesson by summarizing the four steps for solving multi-step inequalities and encouraging viewers to practice these steps. It includes a reminder to apply the steps to all types of inequality symbols. The instructor thanks the viewers for joining the 'Magic of Math' and invites them to subscribe for more educational content. The lesson concludes with a friendly sign-off, wishing viewers a great day and expressing hope for their return to the channel.

Keywords

💡Multi-step Inequalities

Multi-step inequalities are mathematical expressions that involve more than one operation to solve and find the range of values that satisfy the inequality. They are the main focus of the video, as the script aims to teach viewers how to solve these complex inequalities by breaking them down into simpler steps.

💡Distributive Property

The distributive property is a fundamental arithmetic principle that states that the product of a number and a sum is the same as the sum of the products of the addends. In the context of the video, it is used to clear parentheses in an inequality, making it easier to isolate the variable term.

💡Variable Term

A variable term in an inequality is the term that contains the unknown variable, such as 'x'. The script emphasizes the importance of isolating the variable term to one side of the inequality to simplify the expression and solve for the variable.

💡Coefficient

In mathematics, the coefficient is the numerical factor that multiplies the variable in an algebraic term. The video aims to simplify inequalities to a point where the variable has a coefficient of one, making it easier to solve for the variable.

💡Zero Pair

A zero pair in the context of solving inequalities refers to the process of adding or subtracting the same value from both sides of the inequality to eliminate a term and simplify the expression. The script uses this concept to isolate the variable term.

💡Inequality Symbol

Inequality symbols, such as 'greater than', 'less than', 'greater than or equal to', and 'less than or equal to', are used to denote the relationship between two expressions. The video script explains how to handle these symbols when solving multi-step inequalities.

💡Graphing Solution

Graphing the solution of an inequality involves representing the solution set on a number line, which helps visualize the range of values that satisfy the inequality. The video provides step-by-step instructions on how to graph the solution after solving the inequality.

💡Number Line

A number line is a visual tool used in mathematics to represent real numbers in a sequential order. In the video, the number line is used to graph the solutions of inequalities, showing where the variable's value should be included or excluded.

💡Closed Circle

In the context of graphing inequalities, a closed circle indicates that the endpoint value is included in the solution set. The video script explains when to use a closed circle on a number line when graphing the solution of an inequality.

💡Open Circle

An open circle on a number line signifies that the endpoint value is not included in the solution set. The video script demonstrates when to use an open circle when the inequality does not include the endpoint value.

💡Master Math

The phrase 'master math' in the video's title and throughout the script refers to the educational goal of the video—to help viewers gain proficiency in mathematical concepts, specifically in solving multi-step inequalities.

Highlights

Introduction to solving multi-step inequalities using prior knowledge of equations and inequalities.

Four-step process for solving multi-step inequalities: clear parentheses, circle variable terms, isolate the variable, and graph the solution.

Step one involves using the distributive property if necessary to clear parentheses.

Step two focuses on isolating the variable term by adding or subtracting values from both sides of the inequality.

Step three circles the variable and adjusts coefficients to achieve a coefficient of one.

Step four involves graphing the solution on a number line, noting the type of inequality symbol.

Demonstration of solving an inequality with no parentheses and isolating the variable term.

Explanation of creating a zero pair to isolate the variable and the inverse operation of subtraction.

Isolating x by dividing by the coefficient, maintaining the properties of equality.

Graphing the solution with a closed circle on the number line for 'greater than or equal to'.

Solving an inequality with a negative coefficient and reversing the inequality symbol when multiplying by a negative.

The importance of remembering to reverse the inequality symbol when dealing with negative values.

Another example of solving an inequality, emphasizing the process of creating a zero pair and isolating the variable.

Dividing by a negative number and reversing the inequality symbol to find the solution.

Graphing the solution with an open circle on the number line for 'less than'.

Invitation for viewers to pause the video and practice solving an inequality with provided steps.

Review of solving an inequality with distribution, emphasizing the process of zero pairing and isolating the variable.

Final demonstration of solving and graphing an inequality, including dividing by the variable's coefficient.

Conclusion summarizing the four-step process for solving multi-step inequalities.

Transcripts

hi welcome to the magic of math where we

master math one video at a time

today my lesson is on solving multi-step

inequalities

our objective today is just that we will

solve multi-step inequalities but here's

what i would like you thinking about as

i proceed through the lesson today

how can you use what you already know

about solving equations and inequalities

to solve multi-step inequalities

so we're going to draw on your prior

learning

there are four steps to solving

multi-step inequalities step one

if necessary clear parentheses by

performing the distributive property

step two

circle the variable term

and if necessary add or subtract a value

from both sides of the inequality to

isolate the variable term

reminding you that steps one and two may

or may not be necessary

step three circle the variable if all

that's on the left or the right of the

inequality symbol is the variable term

circle the variable

then if necessary multiply or divide a

value from both sides of the inequality

so the variable has a coefficient of one

so this is our end goal we want to know

what the variable is less than greater

than or less than or equal to or greater

than or equal to so the variable needs

to have a coefficient of one

and step four you're going to graph your

solution

noting that this property or these steps

apply to all the inequality symbols less

than greater than less than or equal to

or greater than or equal to

so let's go ahead and dig in and solve a

multi-step inequality so this is called

multi-step because it's going to take

two steps to find the solution set

so the first thing i want to do is there

are no parentheses so i don't need to

distribute but i'm going to isolate this

variable term first so identifying that

my variable term 6x

is being subtracted by 3.

so i need to create a zero pair here so

that i'm left with just 6x on the left

so to do that the inverse of subtract 3

is to add 3 to each side

so this is 0 pair i'm left with 6x

greater than or equal to and 9 plus 3 is

12.

now

i need to get x all by itself i'm going

to isolate x i want a coefficient of 1.

so the inverse of multiply by 6 is to

divide by 6.

what i do to one side of the inequality

i must do to the other to keep the

properties of equality in check

6 divided by 6 is 1 leaving me x

greater than or equal to 12 divided by 6

is 2.

we are ready to graph so i need my

number line i'm going to put my value of

2 on my number line i need a closed

circle because it can be equal to

and it's going to be everything shaded

to the right of two and including two

let's try this one together

again there's no parentheses so i don't

need to distribute so i'm going to

identify my variable term

now i identify that it's being added by

three so i need to create that zero pair

the inverse of add three is to subtract

3 from each side

when i do that i have my variable term

bring down my inequality symbol and 7

subtract 3 is 4.

now i identify what is happening to the

variable

the variable is being divided by

negative four

the inverse of divide by negative four

is to multiply by negative four what i

do to the left i must also do to the

right

now i am multiplying both sides by a

negative value because i am multiplying

both sides by a negative value i have to

remember my rule and reverse the symbol

so it's going to go from greater than 2

less than

so now i'm ready to simplify

negative 4 divided by negative 4 is 1

leaving me x

i already have my symbol

and 4 times negative 4 is negative 16.

there's my solution now we need to graph

i'm going to get my number line i'm

going to put my value negative 16 on my

number line i need an open circle

because it's not equal to

and it's going to be everything shaded

to the left

now it's your turn i would like you to

pause the video now

solve don't forget to graph and come

back when you're done to check your work

welcome back so we're going to identify

our variable term don't forget whatever

operation comes in front of a term

identifies whether it's positive or

negative this is negative 4x

this is positive 3.

so i need to create a zero pair here

this is positive three so i need to

subtract three from each side to create

my zero pair

so bring down the negative four x bring

down the less than or equal to and 27

subtract 3 is 24.

now my variable x i want to create a

coefficient of 1.

so the inverse of multiply by negative

four is to divide both sides by negative

four

i'm dividing by a negative value so i

must make a plan and reverse my

inequality symbol

now i'm ready to simplify

negative 4 divided by negative 4 is 1

giving me just x or 1 x

24 divided by negative 4 is negative 6.

let's graph our solution set

here's my number line

my value negative six on my number line

i need a closed circle because it can be

equal to

and everything shaded to the right

all right your turn again please pause

the video now

solve

graph your solution and come back to

check your work

welcome back

let's go ahead and identify our variable

term

and then it's being added by 13 we're

going to create our zero pair by doing

the inverse and subtract 13 from each

side

this is 0 leaving me 8x

less than or equal to and negative 3 and

negative 13 are negative 16.

now i look to see what's happening to x

x is being multiplied by eight

the inverse of multiplied by eight is to

divide each side by eight

so eight divided by eight is one leaving

the x less than or equal to and negative

sixteen divided by eight

is negative two

we're going to graph our solution we

want negative 2 on our number line we're

going to use a closed circle because it

can be equal to and we're going to shade

to the left

all right here's one that we have to

distribute first

so we're going to three times x and

three times four which is three x plus

twelve

now we're ready to go into step two we

identify our variable term

it's being added by twelve and we want

to create our zero pair so we're going

to subtract 12 from each side

leaving me this is zero 3x greater than

negative 3

9 subtract 12 is negative 3.

identify what's happening to the

variable it's being multiplied by 3 the

inverse of multiplied by 3 is to divide

by 3. what i do to one side i must do to

the other

3 divided by 3 is 1 leaving me x greater

than

negative 1.

our solution on our number line let's

graph our solution set we need a

negative 1 on our number line we want an

open circle because it is not equal to

and it's going to be shaded to the right

now it's your turn i would like you to

pause the video now

solve and graph your solution

welcome back

so we're going to distribute first 7

times x and 7 times negative 2 so that

gives us 7x subtract 14 less than or

equal to negative 21.

we're identifying what's happening to

our variable term and it's being

subtracted by 14. to create my zero pair

here i'm going to add 14 to each side

so this gives us 7x

less than or equal to and negative 21

plus 14 is negative seven

now here's x being multiplied by seven

the inverse would be to divide both

sides by seven

seven divided by seven is one giving me

x less than or equal to

and negative seven divided by seven is

negative one

let's graph our solution set we're going

to plot negative 1 on our number line

we need a closed circle because it can

be less than or equal to

and we're going to shade everything to

the left

and there you have it that is how you

solve multi-step inequalities in four

simple steps i thank you for joining me

today at the magic of math where we

continue to master math one video at a

time i hope you'll come back soon and

subscribe to my channel have a great day