2Step Equations and Inequalities In this lesson, we will learn how to solve 2step equations, 2step inequalities, and practice solving word problems that lead to these 2step equations and inequalities. If you want to review how to translate words into mathematics, go to this lesson. 2Step Equations One of the most important type of equations to solve is called 2step equations. We use the name 2step 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 "2step 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 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 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 Step 2: We undo the multiplication by 5. We divide both sides of the equation by 5.
5x = 25 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 2step method. Step 1: Add 4 to both sides 6x
 4 > 38 Step 2: Divide both sides by 6 6x
> 42 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 2step method. Step 1: Subtract 10 from both sides 3x
+ 10 < 45 Step 2: Divide both sides by 3 3x
< 35 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 2step Equations At this point we are ready to tackle word problems that lead to 2step 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 2step 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 Step 2: We undo the multiplication by 0.05. We divide both sides of the equation by 0.05.
0.05x = 6 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 = 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 2step Inequalities We can also use the 2step 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 2step 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 Step 2: We undo the multiplication by 11. We divide both sides of the equation by 11.
11x < 110 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
