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 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.
(5 points)
a) Let x represent the original number of colored pencils in your pencil box.
Step 1: You share the pencils equally with your friend, so each of you gets x/2 colored pencils.
Step 2: 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 after sharing and adding is 15, so you can write 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
x/2 = 12
Step 5: To isolate x, we can multiply both sides of the equation by 2:
2 * (x/2) = 2 * 12
x = 24
Therefore, the original number of colored pencils in your box was 24.
b) Let y represent the number of colored pencils you would need originally to have at least 20 pencils left in your box.
Step 1: You share the pencils equally with your friend, so each of you would get y/2 colored pencils.
Step 2: You add 3 more colored pencils to your own box, so you would have y/2 + 3 colored pencils.
Step 3: You want to have at least 20 colored pencils left in your box, so you can write the inequality:
y/2 + 3 >= 20
Step 4: To solve for y, we can start by subtracting 3 from both sides of the inequality:
y/2 >= 20 - 3
y/2 >= 17
Step 5: To isolate y, we can multiply both sides of the inequality by 2:
2 * (y/2) >= 2 * 17
y >= 34
Therefore, you would need at least 34 colored pencils originally to have at least 20 pencils left in your box.
can you shorten this
a) The equation is x/2 + 3 = 15.
To solve for x:
1. Subtract 3 from both sides: x/2 = 12
2. Multiply both sides by 2: x = 24
The original number of colored pencils in your box was 24.
b) The inequality is y/2 + 3 >= 20.
To solve for y:
1. Subtract 3 from both sides: y/2 >= 17
2. Multiply both sides by 2: y >= 34
You would need at least 34 colored pencils originally to have at least 20 pencils left in your box.
can you make it seem like an 8th grader did it
a) Let's say you started with some colored pencils, and you shared them equally with your friend. After sharing, you added 3 more colored pencils to your own box. And guess what? You ended up with a total of 15 colored pencils.
So, to find out how many colored pencils you started with, we can use this equation: x/2 + 3 = 15.
To solve for x, we need to follow these steps:
1. Subtract 3 from both sides: x/2 = 12
2. Now, if we want to find the value of x, we need to cancel out the division by 2. We can do that by multiplying both sides by 2: x = 24
So, originally you had 24 colored pencils in your box.
b) Say you wanted to have at least 20 colored pencils left in your box after sharing equally with your friend and adding 3 more colored pencils to your own box.
We can set up this inequality to figure out how many colored pencils you would need to start with: y/2 + 3 >= 20.
To solve for y, we can do these steps:
1. Subtract 3 from both sides: y/2 >= 17
2. To get the value of y, we need to cancel out the division by 2. We can do this by multiplying both sides by 2: y >= 34
So, you would need at least 34 colored pencils originally to have at least 20 pencils left in your box.
a) Let's denote the original number of colored pencils in your box as x.
Step 1: You share them equally with your friend, so each of you will receive x/2 colored pencils.
Step 2: You add 3 more colored pencils to your own box, so you now have x/2 + 3 colored pencils.
Step 3: The total number of colored pencils after sharing and adding is given as 15. Therefore, we can write the equation as:
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
x/2 = 12
Step 5: To isolate x, we can multiply both sides of the equation by 2:
2 * (x/2) = 2 * 12
x = 24
So the original number of pencils in your box was 24.
b) If you wanted to have at least 20 pencils left in your box, you can set up the following inequality:
x/2 + 3 >= 20
Step 1: Start by subtracting 3 from both sides of the inequality:
x/2 >= 20 - 3
x/2 >= 17
Step 2: To isolate x, we can multiply both sides of the inequality by 2:
2 * (x/2) >= 2 * 17
x >= 34
Therefore, you would have needed originally at least 34 colored pencils in your box to have at least 20 pencils left.
c) The inequality we derived in part b can be written as:
x/2 + 3 >= 20
a) Let's break down the problem and write an equation to solve for x:
1. You originally have x colored pencils in your pencil box.
2. You share them equally between you and your friend, so each of you gets x/2 colored pencils.
3. You add 3 more colored pencils to your own box after sharing them, so you have x/2 + 3 colored pencils.
4. This gives you a total of 15 pencils, so the equation becomes: x/2 + 3 = 15.
To solve this equation, we need to isolate x on one side:
x/2 + 3 = 15.
Subtract 3 from both sides:
x/2 = 15 - 3.
Simplify:
x/2 = 12.
Multiply both sides by 2 to get rid of the fraction:
2(x/2) = 2(12).
x = 24.
Therefore, originally, you had 24 colored pencils in your box.
b) If you wanted to have at least 20 pencils left in your box, we can set up an inequality to solve for x:
1. Let's denote the number of pencils left in your box as L.
2. We want to have at least 20 pencils left, so we can write the inequality: L ≥ 20.
3. The number of pencils left in your box can be calculated as the original number of pencils minus the ones you shared and the ones you added later: L = x - (x/2 + 3).
To solve this inequality, we would isolate x:
L = x - (x/2 + 3).
Simplify:
L = x - x/2 - 3.
Multiply both sides by 2 to get rid of the fraction:
2L = 2x - x - 6.
Combine like terms:
2L = x - 6.
Bring all the terms to one side:
x - 2L ≥ 6.
Now let's substitute L with 20:
x - 2(20) ≥ 6.
Simplify:
x - 40 ≥ 6.
Add 40 to both sides:
x ≥ 6 + 40.
x ≥ 46.
Therefore, originally, you would have needed at least 46 colored pencils in your box if you wanted to have at least 20 pencils left.
c) The inequality we used in part b) to solve for x is:
x - 2L ≥ 6.
This inequality represents the condition that you want to have at least 20 pencils left in your box, where L denotes the number of pencils left in your box. The steps to solve for x are as follows:
1. Substitute L with the desired number of pencils left in your box.
2. Simplify the equation.
3. Bring all the terms to one side so that x is on the left side.
4. Solve for x.
In this case, we substituted L with 20 in step 4. The inequality x - 2L ≥ 6 then became x - 2(20) ≥ 6.