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
a. Let's write an equation to solve for x.
First, we can represent the number of colored pencils each of you have after sharing equally as x/2.
After adding 3 more colored pencils to your own box, you have (x/2) + 3 pencils.
Since the total number of pencils you have after sharing and adding is 15, we can write the equation:
(x/2) + 3 = 15
To solve this equation, we can start by subtracting 3 from both sides:
(x/2) = 15 - 3
(x/2) = 12
To isolate x, we can multiply both sides by 2:
x = 12 * 2
x = 24
So, the original number of colored pencils in your box was 24.
b. Let's find out how many colored pencils you would have needed originally to have at least 20 left.
We know that after dividing equally between you and your friend, you would have (x/2) colored pencils.
To have at least 20 pencils remaining, we can write the inequality:
(x/2) >= 20
To solve this inequality, we can multiply both sides by 2 (since we want to isolate x):
x >= 20 * 2
x >= 40
Therefore, you would have needed at least 40 colored pencils originally to have at least 20 left in your box.
c. x >= 40 (where x represents the original number of colored pencils)
What is an inequality for the original equation?
The original equation was:
(x/2) + 3 = 15
To create an inequality from this equation, we can replace the equal sign (=) with a greater than or equal to sign (≥):
(x/2) + 3 ≥ 15
a. Let's solve for x, the original number of colored pencils in your box:
Step 1: You share the colored pencils equally between you and your friend, so each of you gets x/2 colored pencils.
Step 2: After sharing, you add 3 more colored pencils to your own box, so you have x/2 + 3 colored pencils.
Step 3: The total number of pencils you have is 15, so we can set up the equation: x/2 + 3 = 15.
Step 4: To isolate x, we can subtract 3 from both sides of the equation: x/2 = 15 - 3 = 12.
Step 5: Finally, to solve for x, we multiply both sides of the equation by 2: x = 2 * 12 = 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 the inequality as follows:
x/2 + 3 ≥ 20
To solve for x, we isolate it by subtracting 3 from both sides:
x/2 ≥ 20 - 3
x/2 ≥ 17
To further isolate x, we multiply both sides by 2:
x ≥ 17 * 2
x ≥ 34
Therefore, you would have needed at least 34 colored pencils originally to have at least 20 colored pencils left in your box.
c. The inequality for this scenario is:
x/2 + 3 ≥ 20
To solve this inequality, we can subtract 3 from both sides:
x/2 ≥ 20 - 3
x/2 ≥ 17
To further isolate x, we multiply both sides by 2:
2 * (x/2) ≥ 2 * 17
x ≥ 34
Therefore, the solution to the inequality is x ≥ 34.
a. To solve for x, the original number of colored pencils in your box, we can set up the equation:
(x / 2) + 3 = 15
Here's how to solve the equation step by step:
1. Since you shared the colored pencils equally with your friend, you divide the original number of colored pencils by 2: x / 2.
2. After sharing them, you added 3 more colored pencils to your own box, so you add 3 to the divided result: (x / 2) + 3.
3. Finally, you know that this sum equals 15, so you set up the equation: (x / 2) + 3 = 15.
To solve the equation, follow these steps:
1. Subtract 3 from both sides: (x / 2) + 3 - 3 = 15 - 3
Simplifies to: (x / 2) = 12
2. Multiply both sides by 2 to isolate x: 2 * (x / 2) = 12 * 2
Simplifies to: x = 24
Therefore, the original number of colored pencils in your box was 24 (x = 24).
b. To find out how many colored pencils you would have needed originally to have at least 20 left in your box, we can rewrite the equation as an inequality:
(x / 2) + 3 ≥ 20
Here's how to solve the inequality step by step:
1. Subtract 3 from both sides: (x / 2) + 3 - 3 ≥ 20 - 3
Simplifies to: (x / 2) ≥ 17
2. Multiply both sides by 2 to isolate x: 2 * (x / 2) ≥ 17 * 2
Simplifies to: x ≥ 34
Therefore, you would have needed at least 34 colored pencils originally to have at least 20 left in your box.
c. The inequality we set up in part (b) can serve as the solution for part (c):
x ≥ 34