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 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 pencils in your box)
b) what if you wanted to have at least 20 pencils left in your box? how many pencils would you have needed originally>
c.) write an inequality and show all steps for solving for x
a) Let's represent the original number of pencils in the box as x.
Step 1: Sharing equally between you and your friend means you each get x/2 colored pencils.
Step 2: Adding 3 more colored pencils to your box gives you x/2 + 3 pencils.
Step 3: The total number of pencils you have after sharing and adding 3 more is x/2 + 3 + x/2 = 15.
Step 4: Simplifying the equation:
x/2 + x/2 + 3 = 15
2x/2 + 3 = 15
x + 3 = 15
Step 5: Subtracting 3 from both sides of the equation:
x + 3 - 3 = 15 - 3
x = 12
Therefore, the original number of pencils in your box is 12.
b) If you want to have at least 20 pencils left in your box, let's represent the original number of pencils needed as y.
Step 1: Sharing equally between you and your friend means you each get y/2 colored pencils.
Step 2: Adding 3 more colored pencils to your box gives you y/2 + 3 pencils.
Step 3: The total number of pencils you have after sharing and adding 3 more is y/2 + 3 + y/2 = 20.
Step 4: Simplifying the equation:
y/2 + y/2 + 3 = 20
2y/2 + 3 = 20
y + 3 = 20
Step 5: Subtracting 3 from both sides of the equation:
y + 3 - 3 = 20 - 3
y = 17
Therefore, the original number of pencils needed in your box to have at least 20 left is 17.
c) The inequality representing the situation where you want to have at least 20 pencils left in your box is:
y/2 + 3 ≥ 20
Step 1: Subtracting 3 from both sides of the inequality:
y/2 ≥ 17
Step 2: Multiplying all terms by 2 (to get rid of the fraction):
2(y/2) ≥ 2(17)
y ≥ 34
Therefore, the original number of pencils needed in your box is greater than or equal to 34.
a) Let's solve for x, the original number of pencils in your box.
Step 1: You share the pencils equally between you and your friend. So, you have x/2 colored pencils left in your box.
Step 2: You add 3 more colored pencils to your box. So, you have x/2 + 3 colored pencils total.
Step 3: According to the problem, you have a total of 15 colored pencils. So, we can set up the equation: x/2 + 3 = 15.
Step 4: To solve for x, we can start by subtracting 3 from both sides of the equation: x/2 = 15 - 3.
Step 5: Simplifying the right-hand side, we have: x/2 = 12.
Step 6: To isolate x, we multiply both sides of the equation by 2: 2(x/2) = 2(12).
Step 7: Simplifying, we get: x = 24.
Therefore, the original number of colored pencils in your box was 24.
b) Now let's find out how many pencils you would have needed originally to have at least 20 pencils left in your box.
Step 1: Start with the equation we used in part a: x/2 + 3 = 15.
Step 2: We want to have at least 20 colored pencils left, so we can set up the inequality: x/2 + 3 >= 20.
Step 3: Subtracting 3 from both sides of the inequality, we have: x/2 >= 20 - 3.
Step 4: Simplifying, we get: x/2 >= 17.
Step 5: To isolate x, we multiply both sides of the inequality by 2 (since dividing by a fraction is the same as multiplying by its reciprocal): 2(x/2) >= 2(17).
Step 6: Simplifying, we have: x >= 34.
Therefore, you would have needed at least 34 colored pencils originally to have at least 20 pencils left in your box.
a) Let's set up an equation to solve for x, the original number of pencils in your box:
Original number of pencils in your box: x
After sharing them equally with your friend: x/2
Adding 3 more pencils: x/2 + 3
Total number of pencils after adding 3 more: (x/2) + 3
Total number of pencils after sharing and adding = 15
We can now set up the equation:
(x/2) + 3 = 15
First, subtract 3 from both sides:
(x/2) = 15 - 3
(x/2) = 12
Next, multiply both sides by 2 to isolate x:
2 * (x/2) = 2 * 12
x = 24
So the original number of pencils in your box was 24.
b) Now let's figure out how many pencils you would have needed originally to have at least 20 pencils left in your box.
Total number of pencils left in your box: 20
After sharing them equally with your friend: 20/2 = 10
We can set up the equation:
10 + 3 = x
13 = x
So you would have needed at least 13 pencils originally to have 20 pencils left in your box.
c) To write an inequality for the original number of pencils in your box (x) in order to have at least 15 pencils left, we use the expression (x/2 + 3) ≥ 15.
So, (x/2 + 3) ≥ 15.
To solve for x, we can first subtract 3 from both sides:
(x/2) ≥ 15 - 3
(x/2) ≥ 12
Next, multiply both sides by 2:
2 * (x/2) ≥ 2 * 12
x ≥ 24
So the inequality to have at least 15 pencils left in your box is x ≥ 24.