2-Step Equations and Inequalities
In this lesson, we will learn how to solve 2-step equations, 2-step inequalities, and practice solving word problems that lead to these 2-step equations and inequalities. If you want to review how to translate words into mathematics, go to this lesson.
2-Step Equations
One of the most important type of equations to solve is called 2-step equations. We use the name 2-step equation to remind us that there will be a 2 step process involved in solving these equations. We will illustrate this with an example:
Example 1
Solve
3x + 5 = 26
Solution
When the instructions are "solve" we know that our goal is to obtain a result in the form
x = a number
In the equation
3x + 5 = 26
we see that x is not by itself on the left hand side. There are two numbers, 3 and 5, that accompany the variable. Since there are two such numbers, we can expect that it will take one step each to handle these numbers, thus, the name "2-step equations". The left hand side states, "Start with a number, x, multiply this number by 3 and then add 5. The process to get rid of the 3 and the 5 is to undo the multiplication and addition in backwards order. To undo addition, we use subtraction. In particular, we subtract 5 from both sides of the equation.
Step 1: Subtract 5 from both sides:
3x
+ 5 = 26
-5
-5
--------------------------
3x =
21
Next we get rid of the multiplication by 3. To undo multiplication, we divide. In particular, divide both sides of the equation by 3.
Step 2: Divide both sides by 3
3x
= 21
---- ----
3
3
x = 7
At this point, it is always a good idea to check your solution by plugging x back into the original equation. we have
3(7) + 5 = 21 + 5
= 26
Which checks. Since you have plenty of time on the CAHSEE, you will always want to check your answers.
Now try one by yourself. If you want to see the answer, put your mouse on the yellow rectangle and the answer will appear.
Exercise 1
Solve
4x + 3 = 39
Answer
Example 2
Solve
5x - 2 = 23
Solution
We are asked to solve, so we want to end up with "x = a number". The obstacles here are the multiplication by 5 and the subtraction by 2.
Step 1: We first undo subtraction by 2. We add 2 to both sides of the equation.
5x
- 2 = 23
+ 2
+ 2
---------------------------
5x =
25
Step 2: We undo the multiplication by 5. We divide both sides of the equation by 5.
5x = 25
----
----
5
5
x = 5
Now check the solution
5(5) - 2 = 25 - 2
= 23
And the solution checks.
Now try one by yourself. If you want to see the answer, put your mouse on the yellow rectangle and the answer will appear.
Exercise 2
Solve
8x + 1 = 31
Answer
Two Step Inequalities
If the equality is changed to an inequality, then we proceed in the same way, using the two step process.
Example 3
Solve
6x - 4 > 38
Here we have a ">" sign instead of an "=" sign. We can still use the two step method. We see that x is multiplied by 6 and then a 4 is subtracted. We undo these operations using the 2-step method.
Step 1: Add 4 to both sides
6x
- 4 > 38
+ 4
+ 4
-------------------------
6x
> 42
Step 2: Divide both sides by 6
6x
> 42
---- ----
6
6
x > 7
Now try one by yourself. If you want to see the answer, put your mouse on the yellow rectangle and the answer will appear.
Exercise 3
Solve
2x - 7 > 23
Answer
Example 4
Solve
3x + 10 < 45
Solution
We use the two step method. We see that x is multiplied by 3 and then a 10 is added. We undo these operations using the 2-step method.
Step 1: Subtract 10 from both sides
3x
+ 10 < 45
- 10
- 10
-------------------------
3x
< 35
Step 2: Divide both sides by 3
3x
< 35
---- ----
3
3
x < 11 2/3
Now try one by yourself. If you want to see the answer, put your mouse on the yellow rectangle and the answer will appear.
Exercise 4
Solve
5x - 7 < 23
Answer
A Note About Negative Numbers: Although it might not occur on the CAHSEE, it is worth noting what to do if the number in front of the x is negative. It turns out that when we divide or multiply both sides of an inequality by a negative number, we must switch the inequality sign. If you want to see a couple of examples of this click here.
Word Problems and 2-step Equations
At this point we are ready to tackle word problems that lead to 2-step equations. If you want to review how to translate phrases into algebra you can go back to that lesson by clicking here. The best way to become proficient with word problems is to practice. We begin with an example.
Example 5
Chuck's phone company charges a flat fee of $12 per month plus $0.05 per minute. His bill for last month was $18. How many minutes did Chuck talk on the phone last month?
Solution
We will first need to translate the phrases into algebra. Since we want to find the number of minutes that Chuck talked on the phone, we let
x = number of minutes on the phone
The total cost for the month will be
Flat Fee + Cost for Minutes Used = Total Cost
The flat fee is $12. To find the cost for the minutes used, we see that if it is $0.05 per minute and there are x minutes used, then the cost for the minutes used is
Cost for Minutes Used = 0.05x
The total cost was $18. Putting this all together gives
12 + 0.05x = 18
We have now arrived at a 2-step equation. We see that x is multiplied by 0.05 and then 12 is added. We can now solve.
Step 1: We first undo addition by 12. We subtract 12 from both sides of the equation.
12
+ 0.05x = 18
- 12
- 12
---------------------------
0.05x
= 6
Step 2: We undo the multiplication by 0.05. We divide both sides of the equation by 0.05.
0.05x = 6
--------
-------
0.05
0.05
To divide by 0.05 it is easiest to first multiply both the numerator and the denominator by 100 to get rid of the decimal. We have
6(100)
600
--------------
= ------
0.05(100)
5
= 120
We check our answer by plugging 120 back in for x.
12 + 0.05(120) = 12 + 6
= 18
The solution checks out and we can conclude that Chuck used his phone for 120 minutes last month.
Now try one by yourself. If you want to see the answer, put your mouse on the yellow rectangle and the answer will appear.
Exercise 5
Louisa works as a counselor. She meets with each of her clients for 20 minutes and during her 8 hour (480 minute) work day, she takes breaks that total 40 minutes. How many clients can she meet with during her work day?
Answer
Word Problems and 2-step Inequalities
We can also use the 2-step process to solve word problems that can be translated into inequalities. Here is an example.
Example 6
Natasha wants to treat her friends to the movies. She can spend at most $131. The movie tickets cost $11 each and she also wants to spend $21 worth of popcorn and candy for her friends to share. What are the possible number of friends she can invite (including herself)?
Solution
Since she does not have to spend all of her money, this is an inequality. The goal is to find the possible number of friends she can invite, so we label this as x. We set
x = Number of friends that she can invite
Next we look at the cost. We have
Cost of popcorn and candy + Cost of tickets < Total Budget
The cost of the popcorn and candy is $21. Since the tickets are $11 each and she is inviting x friends, the cost of the tickets is
Cost of tickets = 11x
The total budget is $121. Putting this all together gives
21 + 11x < 131
We have now arrived at a 2-step inequality. We see that x is multiplied by 11 and then 21 is added. We can now solve.
Step 1: We first undo addition by 21. We subtract 21 from both sides of the equation.
21
+ 11x < 131
- 21
- 21
---------------------------
11x
< 110
Step 2: We undo the multiplication by 11. We divide both sides of the equation by 11.
11x < 110
------
------
11
11
We have
x < 10
We can conclude that Natasha can invite at most 10 of her friends.
Now try one by yourself. If you want to see the answer, put your mouse on the yellow rectangle and the answer will appear.
Exercise 6
An office building contains 5,100 square feet of space. Each employee will have a cubicle that takes up 200 square feet of space. The plan is to have a 300 square foot entryway. What are the possible number of employees the office building can accommodate?
Answer