16. You originally have x colored pencils in your pencil box. You share them equally between you and your friend. You add in 3 more colored pencils to your own box after sharing them. This gives you a total of 15 pencils.
a. Write an equation and show all of your steps for solving x (the original number of colored pencils in your box).
b. If you wanted to have at least 20 colored pencils left in your box, how many colored pencils would you have needed originally?
c. Write an inequality and show all steps for solving for x.
(5 points)
a. To find the original number of colored pencils, we can set up the equation:
(x/2) + 3 = 15
First, we divide x by 2, since we are sharing the pencils equally. Then, we add 3 to account for the additional colored pencils. Finally, we set this equal to 15, since that is the total number of pencils after sharing and adding 3.
To solve for x, we can subtract 3 from both sides of the equation:
(x/2) = 12
Then, we can multiply both sides of the equation by 2 to isolate x:
x = 24
So, the original number of colored pencils in your box was 24.
b. If we wanted to have at least 20 colored pencils left in your box, we can set up the inequality:
(x/2) + 3 >= 20
First, we divide x by 2, since we are sharing the pencils equally. Then, we add 3 to account for the additional colored pencils. Finally, we set this greater than or equal to 20, since that is the minimum number of pencils we want to have left in the box.
To solve for x, we can subtract 3 from both sides of the inequality:
(x/2) >= 17
Then, we can multiply both sides of the inequality by 2 to isolate x:
x >= 34
Therefore, the original number of colored pencils you would have needed would be at least 34.
c. The inequality to solve for x would be:
(x/2) + 3 >= 20
Following the same steps as in part b, we can solve for x:
(x/2) >= 17
x >= 34
So, the inequality to solve for x is x >= 34.
a. Let's set up the equation to solve for x, which represents the original number of colored pencils in your box.
Step 1: Sharing them equally between you and your friend means dividing the pencils by 2.
So, you would have half of the original number of colored pencils: x/2.
Step 2: Adding in 3 more colored pencils to your own box after sharing them: x/2 + 3.
Step 3: The total number of colored pencils after adding 3 and sharing equally is 15.
So, the equation becomes: x/2 + 3 = 15.
To solve for x, we need to isolate x by following these steps:
Step 4: Subtract 3 from both sides of the equation: x/2 = 12.
Step 5: Multiply both sides of the equation by 2 to remove the fraction: 2 * (x/2) = 2 * 12.
Step 6: Simplify: x = 24.
Therefore, the original number of colored pencils in your box was 24.
b. If you wanted to have at least 20 colored pencils left in your box, we can set up an inequality to solve for the original number of colored pencils (x).
The equation would be: (x/2) + 3 ≥ 20.
To solve for x, we follow these steps:
Step 1: Subtract 3 from both sides of the inequality: (x/2) ≥ 17.
Step 2: Multiply both sides of the inequality by 2 to remove the fraction: 2 * (x/2) ≥ 2 * 17.
Step 3: Simplify: x ≥ 34.
Therefore, you would have needed at least 34 colored pencils in your box originally to have at least 20 left after sharing them equally and adding 3 more.
c. The inequality to solve for x would be:
(x/2) + 3 ≥ 20.
To solve for x, we follow the same steps as in part b:
Step 1: Subtract 3 from both sides of the inequality: (x/2) ≥ 17.
Step 2: Multiply both sides of the inequality by 2 to remove the fraction: 2 * (x/2) ≥ 2 * 17.
Step 3: Simplify: x ≥ 34.
Therefore, the inequality solution is x ≥ 34. This means that the original number of colored pencils in your box should be greater than or equal to 34 to have at least 20 left after sharing and adding 3 more.
a. Let's solve for x using an equation:
Step 1: Let's assume the number of colored pencils you originally had in your box is x.
Step 2: When you share them equally with your friend, you both get x/2 pencils.
Step 3: After sharing, you add 3 more pencils to your own box, so you have x/2 + 3 pencils.
Step 4: The total number of pencils is 15 (according to the problem).
Step 5: Set up the equation: x/2 + 3 = 15.
Step 6: Subtract 3 from both sides to isolate x: x/2 = 12.
Step 7: Multiply both sides by 2 to solve for x: x = 24.
Therefore, the original number of colored pencils in your box was 24.
b. Let's solve for the number of pencils needed originally to have at least 20 left in your box:
Step 1: Let's assume the number of colored pencils you originally had in your box is x.
Step 2: You share them equally with your friend, so you both get x/2 pencils.
Step 3: After sharing, you have x/2 pencils in your box.
Step 4: In order to have at least 20 colored pencils left, we set up the inequality: x/2 ≥ 20.
Step 5: Multiply both sides by 2 to get rid of the fraction: x ≥ 40.
Therefore, you would have needed at least 40 colored pencils originally in your box to have at least 20 colored pencils left.
c. The inequality for solving x is: x/2 ≥ 20.
By multiplying both sides by 2, we get: x ≥ 40.
Therefore, the solutions to the inequality are all values of x greater than or equal to 40.