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's solve for x, the original number of pencils in your box.
Step 1: Share the pencils equally with your friend. This means you divide the pencils by 2.
Step 2: After sharing equally, there are x/2 pencils left in your box.
Step 3: Add 3 more colored pencils to your own box.
Step 4: The total number of pencils in your box is now x/2 + 3.
Step 5: According to the problem, this total is 15. So we set up the equation:
x/2 + 3 = 15
Step 6: Subtract 3 from both sides of the equation to isolate x/2.
x/2 = 15 - 3
x/2 = 12
Step 7: Multiply both sides of the equation by 2 to solve for x.
(x/2) * 2 = 12 * 2
x = 24
Therefore, the original number of pencils in your box was 24.
b) To solve for the original number of pencils if you want to have at least 20 pencils left in your box:
Step 1: After sharing equally with your friend, you will have x/2 pencils left in your box.
Step 2: You add an additional 3 colored pencils to your box, so the total number of pencils will be x/2 + 3.
Step 3: According to the problem, you want to have at least 20 pencils left in your box, so we set up the inequality:
x/2 + 3 ≥ 20
Step 4: Subtract 3 from both sides of the inequality to isolate x/2.
x/2 ≥ 20 - 3
x/2 ≥ 17
Step 5: Multiply both sides of the inequality by 2 to solve for x.
(x/2) * 2 ≥ 17 * 2
x ≥ 34
Therefore, you would have needed originally at least 34 pencils in your box to have at least 20 left.
c) The inequality is x/2 + 3 ≥ 20.
a) Let's solve for x using the given information.
Step 1: Start with the original number of pencils, x.
Step 2: Share them equally between you and your friend. Each of you would receive x/2 pencils.
Step 3: Add 3 more pencils to your own box. This gives you x/2 + 3 pencils.
Step 4: This total is now 15 pencils. Write the equation: x/2 + 3 = 15.
Step 5: Subtract 3 from both sides of the equation to isolate x/2. This gives x/2 = 12.
Step 6: Multiply both sides of the equation by 2 to solve for x. This gives x = 24.
Therefore, the original number of pencils in your box was 24.
b) To find out how many pencils you would have needed originally to have at least 20 left, let's set up an equation.
Step 1: Start with the original number of pencils, x.
Step 2: Share them equally between you and your friend. Each of you would receive x/2 pencils.
Step 3: Add 3 more pencils to your own box. This gives you x/2 + 3 pencils.
Step 4: Subtract this total from the required minimum of 20 pencils and set it equal to zero. Write the equation: 20 - (x/2 + 3) = 0.
Step 5: Simplify the equation. 20 - (x/2 + 3) = 0 becomes 20 - x/2 - 3 = 0.
Step 6: Combine like terms. 17 - x/2 = 0.
Step 7: Multiply both sides of the equation by 2 to eliminate the fraction. This gives 34 - x = 0.
Step 8: Add x to both sides and subtract 34 from both sides of the equation to solve for x. This gives x = 34.
Therefore, you would have needed to have initially at least 34 pencils in your box.
c) To write an inequality and solve for x, let's consider the scenario where you want to have at least 15 pencils left in your box.
Step 1: Start with the original number of pencils, x.
Step 2: Share them equally between you and your friend. Each of you would receive x/2 pencils.
Step 3: Add 3 more pencils to your own box. This gives you x/2 + 3 pencils.
Step 4: Subtract this total from the minimum requirement of 15 pencils and write the inequality: (x/2 + 3) ≥ 15.
Step 5: Subtract 3 from both sides of the inequality. This gives x/2 ≥ 12.
Step 6: Multiply both sides of the inequality by 2 to eliminate the fraction. This gives x ≥ 24.
Therefore, for you to have at least 15 pencils left in your box originally, you would have needed to have at least 24 pencils.