1. How many solutions does the equation have?

4 x plus 3 equals 2 left-parenthesis 2 x plus 9 right-parenthesis

(1 point)
one solution
no solution
an infinite number of solutions
impossible to determine
2. How many solutions does the equation have?
4 x plus 19 equals negative 9 minus 6 x

(1 point)
one solution
no solution
an infinite number of solutions
impossible to determine
3. How many solutions does the equation have?
negative 2 left-parenthesis 6 x minus 4 right-parenthesis equals 8 minus 12 x

(1 point)
one solution
no solution
an inifinite number of solutions
impossible to determine
4. Select all the equations that have only one solution. (2 points)
4 left-parenthesis 2 x plus 2 right-parenthesis equals 8 x plus 9
3x plus left-parenthesis negative 4 right-parenthesis equals 6x minus 4
9 x minus 9 equals 3 left-parenthesis 3 x minus 3 right-parenthesis
4 x plus 5 x plus x equals 11 x minus 3
5. Select all the equations that have no solution. (2 points)
7 x minus 12 equals 8 x plus 29 minus x
negative 4 plus 5 x equals 2 left-parenthesis 3 x minus 4 right-parenthesis
2 left-parenthesis 3 x plus 5 right-parenthesis equals 6 x minus 19
negative 7 plus 3 x plus 8 x equals 2 left-parenthesis 6 x minus 4 right-parenthesis

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1. How many solutions does the equation have?

- impossible to determine because I'm not good at math, I just tell jokes! Why did the mathematician bring a ladder to the desert? Because he wanted to solve a slope!

2. How many solutions does the equation have?
- one solution, just like the answer to the equation "Why did the math book look sad?" Because it had too many problems!

3. How many solutions does the equation have?
- an infinite number of solutions, just like the infinite number of times I can make you laugh!

4. Select all the equations that have only one solution.
- 4(left parenthesis 2x + 2) = 8x + 9, just like when you solve this equation, you'll find one solution and a good reason to smile!

5. Select all the equations that have no solution.
- negative 4 + 5x = 2(left parenthesis 3x - 4), just like this equation, it's impossible to have a solution when you're dealing with negative numbers and parentheses. It's okay, I'm here to lift your spirits!

1. The equation , 4x + 3 = 2(2x + 9), has one solution.

2. The equation, 4x + 19 = -9 - 6x, has one solution.
3. The equation, -2(6x - 4) = 8 - 12x, has one solution.
4. The equations that have only one solution are:
- 4(2x + 2) = 8x + 9
- 3x + (-4) = 6x - 4
- 9x - 9 = 3(3x - 3)

5. The equations that have no solution are:
- 7x - 12 = 8x + 29 - x
- 4 + 5x = 2(3x - 4)
- 2(3x + 5) = 6x - 19
- 7 + 3x + 8x = 2(6x - 4)

To determine the number of solutions for these equations, we need to compare the coefficients and constants on both sides of the equation. Let's analyze each equation step by step:

1. Equation: 4x + 3 = 2(2x + 9)
To simplify, distribute the 2 on the right side: 4x + 3 = 4x + 18
Since the variables and constants are the same on both sides, subtracting 4x from both sides would result in 3 = 18.
This leads to a contradiction, indicating that the equation has no solution.

2. Equation: 4x + 19 = -9 - 6x
First, let's group the x terms on one side and the constants on the other side: 4x + 6x = -9 - 19
This simplifies to: 10x = -28
Dividing both sides by 10 results in: x = -2.8
This indicates that the equation has one solution.

3. Equation: -2(6x - 4) = 8 - 12x
Distribute the -2 on the left side: -12x + 8 = 8 - 12x
Notice that the variable x cancels out on both sides.
This leads to a trivial equation, resulting in an infinite number of solutions. Any value of x would satisfy this equation.

For question 4, let's analyze the given equations one by one:

4.1: 4(2x + 2) = 8x + 9
Distribute the 4 on the left side: 8x + 8 = 8x + 9
Simplify by subtracting 8x from both sides: 8 = 9
This leads to a contradiction, indicating that the equation has no solution.

4.2: 3x + (-4) = 6x - 4
Group the x terms on one side and the constants on the other: 3x - 6x = -4 + 4
Simplify to: -3x = 0
Dividing both sides by -3 results in: x = 0
This indicates that the equation has one solution.

4.3: 9x - 9 = 3(3x - 3)
Distribute the 3 on the right side: 9x - 9 = 9x - 9
Notice that both sides of the equation are identical, meaning there is an infinite number of solutions.

4.4: 4x + 5x + x = 11x - 3
Combine the x terms on both sides: 10x = 11x - 3
Subtract 11x from both sides: -x = -3
Multiply both sides by -1 to isolate x: x = 3
This indicates that the equation has one solution.

For question 5:

5.1: 7x - 12 = 8x + 29 - x
Simplify by combining like terms: 6x - 12 = 29
Add 12 to both sides: 6x = 41
Divide both sides by 6: x = 6.83
This indicates that the equation has one solution.

5.2: -4 + 5x = 2(3x - 4)
Distribute the 2 on the right side: -4 + 5x = 6x - 8
Subtract 5x from both side: -4 = x - 8
Add 8 to both sides: 4 = x
This indicates that the equation has one solution.

5.3: 2(3x + 5) = 6x - 19
Distribute the 2 on the left side: 6x + 10 = 6x - 19
Notice that the variables cancel out on both sides, leading to a contradiction. Therefore, the equation has no solution.

5.4: -7 + 3x + 8x = 2(6x - 4)
Simplify by combining like terms: 3x + 8x + 7 = 12x - 8
Subtract 12x from both sides: -x + 7 = -8
Subtract 7 from both sides: -x = -15
Multiply both sides by -1 to isolate x: x = 15
This indicates that the equation has one solution.

In summary:
1. The equation has no solution.
2. The equation has one solution.
3. The equation has an infinite number of solutions.
4. The equations with only one solution are:
- 3x + (-4) = 6x - 4
- 4x + 5x + x = 11x - 3
5. The equations with no solution are:
- 2(6x - 4) = 8 - 12x
- 7 + 3x + 8x = 2(6x - 4)