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Table of Contents
In this lesson, we will understand how to use variables to represent numbers, how to solve equations with one variable missing, and how to write inequalities to indicate constraints or conditions.
See the fact file below for more information on the one-variable equations and inequalities or alternatively, you can download our 35-page One-Variable Equations and Inequalities worksheet pack to utilise within the classroom or home environment.
Key Facts & Information
VARIABLES
- For this section, we will use the equation below.
- 3 + x = 13
- Notice that the equation is not purely made up of numbers. We have a letter present in the equation, which is x.
- The “x” is the variable.
- VARIABLES – It is a symbol for a number that is missing.
- Variables are not only used in addition equations; they can also be used in multiplication equations like the one presented below.
- 3x = 15
- As in the previous example, the variable in this equation is x.
- Variables are not only used in equations; they can also be used in inequalities.
- y < 2
- The inequality above has one variable, which is y.
- According to this inequality, y is less than 2. Therefore, if we want to solve for this, the value of y is all the numbers less than 2.
SOLVING EQUATIONS WITH ONE VARIABLE
- For this section, we will use the same equation that we used in the previous section.
- 3 + x = 13
- To solve this equation, we have to find the value of x.
- If we want to solve for the value of x, we have to first isolate the variable on one side, which will give us the equation below.
- x = 13 – 3
- Note that since this is an addition equation, if we isolate the variable on one side, we have to remove the number on that side.
- In this case, if we want to isolate the variable on the left side, we have to remove the number 3 on the left side.
- Thus, if we isolate x on the left side, we need to remove 3 on the left side. But note that this is an equation, therefore whatever you do on the left side must also be done on the right side.
- If we want to remove 3 on the left side, we have to subtract 3 from the right side too.
- 3 + x – 3 = 13 – 3
- Now, we can already isolate x on the left side since we are subtracting 3 from 3, giving us 0.
- x + 0 = 13 – 3
- To simplify this, we will get the equation below.
- x = 13 – 3
- Solving for the value of x, we need to solve 13 minus 3.
- x = 10
- Therefore, the value of x is 10.
- To check if we got the correct answer, we can substitute the value that we got for x to the original equation that we used.
- 3 + x = 13
- Replacing x with 10 would give us 3 plus 10, and we know that if we add 3 to 10, we will get 13. Therefore, the answer that we got is correct.
- In the previous example, we solved an equation with one variable in the form of:
- x + p = q
- wherein p and q are both non-negative rational numbers.
- Now, let us try solving for an equation, still with one variable, but in the form of:
- x – p = q
- For this form, we will use the equation below.
- x – 5 = 12
- As we did in the previous equation, we have to isolate the variable on one side. In this case, we will be isolating the variable x on the left side.
- x – 5 + 5 = 12 + 5
- As before, since we are solving for a subtraction equation, we have to add the number that we have to remove in order to get the value 0.
- In this case, since we have to remove 5 from the left side, we have to add 5 on both sides.
- x – 0 = 12 + 5
- Since we are just subtracting 0 from x on the left side, we can already remove it, giving us:
- x = 12 + 5
- Given this equation, we can now solve for the value of x.
- To solve for the value of x, we have to find the sum of 12 and 5.
- 12 + 5 = 17
- We know that if we add 12 and 5 together, we will get 17.
- Therefore, the value of x is equal to 17.
- x = 17
- As before, to check if we got the right answer, we can replace x in the original equation with the value that we got, which is 17.
- 17 – 5 = 12
- We know that if we subtract 5 from 17 it would give us 12. Thus, the answer that we got is correct.
- Now that we have tried both addition equations and subtraction equations, we will move on to the next form, which is:
- px = q
- where both p and q are non-negative rational numbers.
- For this form, we will use the sample equation below.
- 3x = 12
- As before, we have to isolate x on one side.
- To isolate x on the left side, we have to remove the number that is attached to it. In this case, we have to remove 3.
- To remove 3 on the left side, we have to divide both sides by 3.
- Like before, we have to divide both sides by 3 because this is an equation, therefore whatever you do on the left side must also be done on the right side.
- Now, we know that if we divide a by a, the answer would be 1.
- Therefore, the number that will be attached to x after dividing both sides by 3 will be 1.
- But we know that if we multiply a number by 1, the answer would be the number itself also, which means that we can already remove 1 from the equation in order to avoid confusion.
- The variable x is now isolated, and we can already solve for its value.
- To solve for the value of x we have to perform 12 divided by 3.
- And we know that 12 divided by 3 is equal to 4.
- Therefore, the value of x is 4.
- x = 4
- To check, we can replace the value of x in the original equation with the value that we got, which is 4.
- 3(4) = 12
- We know that if we multiply 3 by 4, the answer that we will get is indeed 12. Thus, the value of x that we got is correct.
- Now, let us try solving an equation in the form:
- This time, we need to multiply both sides by 3 in order to remove 3 from the left side.
- Since we have multiplied both sides by 3, we can already remove the 3 on the left side.
- Removing 3 on the left side would give us:
- x = 6 x 3
- Now, we can solve for the value of x.
- What we need to do in order to solve for the value of x is to multiply 6 with 3.
- 6 x 3 = 18
- If we multiply 6 with 3, the value that we will get is 18.
- Therefore, the value of x is 18.
- x = 18
- To check, we can replace x in the original equation with the value that we got, which is 18.
- We know that if we divide 18 into 3 equal parts, we will get 6.
- Thus, the value of x that we got is correct.
USING VARIABLES IN INEQUALITIES
- In this section, we will use variables in identifying or writing inequalities.
- y > 10
- Looking at the inequality presented, we have one variable, which is y.
- With this, we know that y’s value is greater than 10.
- Following the number line, all the numbers on the right side of 10 can be the value of y.
- 10 > y
- On the other hand, the inequality presented above is telling us that the value of y can be any number that is less than 10.
- 10 > y AND 6 < y
- Here, we have the word “AND” which means that the value of y must be within these constraints defined by the inequalities.
- To better visualize this, we will use a number line.
- We have to first identify the location of the numbers defined, in this case 10 and 6.
- Now, let us look at the inequalities identified.
- The first inequality is 10 > y, therefore the value of y must be less than 10. To visualize this, the value of y must be on the left side of 10.
- Now, let us look at the other inequality.
- The other inequality is 6 < y or y > 6, thus the value of y must be greater than 6 or on the right of 6.
- We have now identified the range of possible values of y.
- Next thing we have to do is to identify the portion where they overlap.
- Thus, the possible values of y are 7, 8, and 9.
- Another way to express the inequalities we have defined previously is to combine them.
- 10 > y AND 6 < y
- 6 < y < 10
- Both ways of expressing the inequalities above mean the same. However, note that the connector that we used in the first expression is AND, which is why we were able to combine the two inequalities into just one.
One-Variable Equations and Inequalities Worksheets
This is a fantastic bundle which includes everything you need to know about the one-variable equations and inequalities across 35 in-depth pages. These are ready-to-use One-Variable Equations and Inequalities worksheets that are perfect for teaching students how to use variables to represent numbers, how to solve equations with one variable missing, and how to write inequalities to indicate constraints or conditions.
Complete List Of Included Worksheets
- Lesson Plan
- One-Variable Equations and Inequalities
- Isolate
- Find X
- Minus
- Fill
- Box
- Direction
- Combine
- Yes or No?
- Possible?
- Identify
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