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 first set up the equation using the given information:
(x/2) + 3 = 15
First, divide x by 2 to find the number of pencils each of you get:
x/2 + 3 = 15
Next, subtract 3 from both sides of the equation:
x/2 = 12
Finally, multiply both sides of the equation by 2 to solve for x:
x = 24
b. If you wanted to have at least 20 colored pencils left in your box, you would set up the following equation:
(x/2) + 3 = 20
First, subtract 3 from both sides of the equation:
x/2 = 17
Next, multiply both sides of the equation by 2 to solve for x:
x = 34
So, you would have needed originally 34 colored pencils.
c. To write an inequality to solve for x, we can use the fact that you added 3 colored pencils after sharing them equally. So the inequality would be:
(x/2) + 3 ≥ 20
First, subtract 3 from both sides of the inequality:
(x/2) ≥ 17
Next, multiply both sides of the inequality by 2 to solve for x:
x ≥ 34
So the original number of colored pencils you must have is at least 34.
Solve the equation filling in the steps: 2(x−3)+21=−3
(6 points)
Distribute to get ______
Combine like terms to get _______
Use the (choose a property) ______ to get (resulting equation) Response area
Use the (choose a property)_________ to get (resulting answer) Response area
Distribute to get: 2x - 6 + 21 = -3
Combine like terms to get: 2x + 15 = -3
Use the subtraction property of equality to get: 2x = -18
Use the division property of equality to get: x = -9
So the resulting answer is x = -9
ax−44<15
To solve the inequality ax - 44 < 15, we will isolate the variable x.
Add 44 to both sides of the inequality:
ax - 44 + 44 < 15 + 44
ax < 59
Divide both sides of the inequality by a (assuming a is positive):
(ax)/a < 59/a
x < 59/a
So the solution to the inequality is x < 59/a, where a is a positive constant.
ax+15=44
To solve the equation ax + 15 = 44, we will isolate the variable x.
Subtract 15 from both sides of the equation:
ax + 15 - 15 = 44 - 15
ax = 29
Divide both sides of the equation by a (assuming a is not equal to 0):
(ax)/a = 29/a
x = 29/a
So the solution to the equation is x = 29/a, where a is a non-zero constant.
he base of a triangle is four less than three times the height. The area of the triangle is 38 square centimeters. A=(b×h)/2
(3 points)
An expression for the base of the triangle: Response area
Equation for the area of the triangle, substituting in for the base: Response area
The first step in solving for h: Response area
An expression for the base of the triangle:
Let h be the height of the triangle. Then the base can be expressed as 3h - 4, since the base is four less than three times the height.
Equation for the area of the triangle, substituting in for the base:
The area of the triangle is given by the formula A = (b × h) / 2. Substituting in the expression for the base, we have:
A = ((3h - 4) × h) / 2
The first step in solving for h:
To solve for h, we can set the equation for the area equal to 38, since we know the area of the triangle is 38 square centimeters:
((3h - 4) × h) / 2 = 38
a. To solve for the original number of colored pencils x, we can set up the equation based on the given information:
(x / 2) + 3 = 15
First, we divide the original number of colored pencils x by 2 to represent sharing them equally between you and your friend. Then, we add 3 to that result to account for the additional colored pencils in your own box. Finally, we set the sum equal to the total number of colored pencils, which is 15.
Now let's solve the equation step by step:
(x / 2) + 3 = 15
Subtract 3 from both sides to isolate the fraction on the left side:
(x / 2) = 15 - 3
Simplify the right side:
(x / 2) = 12
Multiply both sides by 2 to eliminate the fraction:
x = 12 * 2
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 the equation:
(x / 2) + 3 = 20
Following the same steps as before, let's solve the equation:
(x / 2) + 3 = 20
Subtract 3 from both sides:
(x / 2) = 20 - 3
Simplify the right side:
(x / 2) = 17
Multiply both sides by 2:
x = 17 * 2
x = 34
Therefore, you would have needed originally at least 34 colored pencils in your box to have at least 20 left after sharing them equally with your friend.
c. To write the inequality and solve for x, we can set up the expression based on the given condition:
(x / 2) + 3 ≥ 20
Following the same steps as before, let's solve the expression:
(x / 2) + 3 ≥ 20
Subtract 3 from both sides:
(x / 2) ≥ 20 - 3
Simplify the right side:
(x / 2) ≥ 17
Multiply both sides by 2 to eliminate the fraction while preserving the inequality direction:
x ≥ 17 * 2
x ≥ 34
Therefore, the original number of colored pencils in your box should be greater than or equal to 34 to ensure you have at least 20 colored pencils left after sharing them equally with your friend.