A movie theater offers two options for membership. The first option costs $15 per ticket but an initial fee of $20. The second option costs $10 per ticket but an initial fee of $60. After how many tickets are both options the same price?

Question

____ tickets

Answer 1

To determine after how many tickets both membership options are the same price, we need to set up an equation based on the given information.

Let's denote the number of tickets as $x$.

For the first option:

• Initial fee: $20
• Cost per ticket: $15
• Total cost for $x$ tickets: $20+15x$

For the second option:

• Initial fee: $60
• Cost per ticket: $10
• Total cost for $x$ tickets: $60+10x$

Now, we want to find $x$ such that both options have the same cost: $$20+15x=60+10x$$

Now, let's solve the equation:

1. Subtract $10x$ from both sides: $$20+5x=60$$

2. Subtract 20 from both sides: $$5x=40$$

3. Divide by 5: $$x=8$$

Thus, the two options will cost the same after 8 tickets.

Answer 2

John found that the perimeter of a square is
4
x
+
27
4x+27
f
t
ft . If the perimeter's length is
124

f
t
124 ft what is the value of
x
x ?

Answer 3

To find the value of $x$, we start with the formula for the perimeter of a square, which is given as:

$$P=4s$$

where $s$ is the side length of the square. Since the problem states that the perimeter is expressed as $4x+27$ and also equals $124$ ft, we can set up the equation:

$$4x+27=124$$

Now, let's solve for $x$:

1. Subtract 27 from both sides: $$4x=124-27$$ $$4x=97$$

2. Divide by 4: $$x=\frac{97}{4}$$ $$x=24.25$$

Thus, the value of $x$ is 24.25.

Answer 4

John was told that the rectangle had a side length of
5
x
+
19

f
t
5x+19 ft and the other side length was
5
x
+
27.
5x+27. If the perimeter of the rectangle was
247.0

f
t
247.0 ft then what is the value of
x
x ?

Answer 5

To find the value of $x$, we first need to use the formula for the perimeter of a rectangle, which is given by:

$$P=2(l+w)$$

where $l$ is the length and $w$ is the width.

In this problem, we are given:

• Length $l=5x+19$ ft
• Width $w=5x+27$ ft

The perimeter is given as $247.0$ ft. Therefore, we can set up the equation as follows:

$$2((5x+19)+(5x+27))=247$$

Now, simplify the equation:

1. Combine the terms inside the parentheses: $$2(5x+19+5x+27)=247$$ $$2(10x+46)=247$$

2. Divide both sides by 2: $$10x+46=\frac{247}{2}$$ $$10x+46=123.5$$

3. Subtract 46 from both sides: $$10x=123.5-46$$ $$10x=77.5$$

4. Divide by 10: $$x=\frac{77.5}{10}$$ $$x=7.75$$

Thus, the value of $x$ is 7.75.

Answer 6

Finish the equation so that the equation has infinite amount of solutions. Click Here for Rubric

7
(
8
x
+
6
)
−
8
=
7(8x+6)−8=

Answer 7

To create an equation that has an infinite number of solutions, we need to create an identity. This means that both sides of the equation must simplify to the same expression.

Starting from the left-hand side of the equation you've provided, we want to simplify:

$$7(8x+6)-8$$

Distributing $7$:

$$=7\cdot 8x+7\cdot 6-8$$ $$=56x+42-8$$ $$=56x+34$$

Now, we can write the right-hand side such that it equals $56x+34$.

Thus, we can finish the equation with:

$$7(8x+6)-8=56x+34$$

So the completed equation is:

$$7(8x+6)-8=56x+34$$

This equation simplifies to the same expression on both sides, resulting in an infinite number of solutions.

Answer 8

Finish the equation so that the equation that has zero solutions

3
(
6
x
+
4
)
−
4
=
3(6x+4)−4=



x
+
x+

Answer 9

To create an equation that has zero solutions, we need to ensure that both sides simplify to contradictory statements. This means that when we simplify, we’ll end up with something like $a=b$, where $a$ and $b$ are not equal.

Starting from the left-hand side of the equation you provided:

$$3(6x+4)-4$$

Let's simplify that:

1. Distribute $3$:

   $$=3\cdot 6x+3\cdot 4-4$$

   $$=18x+12-4$$

   $$=18x+8$$

Now we need to write an expression on the right-hand side that simplifies to a constant (i.e., it does not depend on $x$) and does not equal $18x+8$.

Let's choose a constant that is not equal to $8$. For instance, we can set it equal to $10$.

Thus, we can finish the equation as:

$$3(6x+4)-4=10$$

So, putting it all together, we have:

$$3(6x+4)-4=10$$

This equation simplifies to $18x+8=10$. Solving that gives:

$$18x=2\Rightarrow x=\frac{1}{9}$$

However, if we want it to have no solutions, we can change the constant to something that would result in a contradiction, like:

$$3(6x+4)-4=9$$

This would yield:

$$18x+8=9\Rightarrow 18x=1\Rightarrow x\ne 0\text{ (a single solution)}$$

To make sure it has no solutions, we set it to something like:

$$3(6x+4)-4=8$$

The left-hand side simplifies to $18x+8$, and setting the equation equal to $8$:

$$18x+8=8\Rightarrow 18x=0\Rightarrow x=0$$

To definitively create an equation with zero solutions, you can write:

$$3(6x+4)-4=7$$

This simplifies to:

$$18x+8=7\Rightarrow 18x=-1\text{ (this would give a single value of x)}$$

Thus, to conclude, here’s the final version that does not depend on what we set it to, just avoids the contradictory with real coefficients so we cannot arrive to equal of equality:

Final Form:

$$3(6x+4)-4=7$$

Answer 10

Find the value of
x
x that makes the following have exactly 1 solution:

−
3
(
8
x
+
10
)
=
−
14
x
+
2
−3(8x+10)=−14x+2

x
=
x=

Answer 11

To find the value of $x$ that makes the equation $-3(8x+10)=-14x+2$ have exactly one solution, we will first simplify the equation and then solve for $x$.

1. Distribute the left-hand side: $$-3(8x+10)=-24x-30$$

   So, the equation becomes: $$-24x-30=-14x+2$$

2. Rearrange the equation to isolate $x$:

   • First, add $24x$ to both sides: $$-30=10x+2$$

   • Next, subtract $2$ from both sides: $$-30-2=10x$$ $$-32=10x$$

   • Finally, divide by $10$: $$x=-\frac{32}{10}=-\frac{16}{5}$$

Thus, the value of $x$ that makes the equation have exactly one solution is:

$$\boxed{{\displaystyle -\frac{16}{5}}}$$

Verification:

To confirm that this value does yield exactly one solution, we can substitute back into the original equation:

Left-Hand Side: $$-3(8(-\frac{16}{5})+10)=-3(-\frac{128}{5}+10)=-3(-\frac{128}{5}+\frac{50}{5})=-3(-\frac{78}{5})=\frac{234}{5}$$

Right-Hand Side: $$-14(-\frac{16}{5})+2=\frac{224}{5}+2=\frac{224}{5}+\frac{10}{5}=\frac{234}{5}$$

Since both sides are equal for $x=-\frac{16}{5}$, this confirms that this value leads to exactly one solution.