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
1. To determine the number of solutions for the equation 4x + 3 = 2(2x + 9), we need to simplify it first.
By distributing the 2, the equation becomes:
4x + 3 = 4x + 18.
Notice that the variables cancel out in this equation, leaving us with 3 = 18.
Since this is a contradiction, there are no solutions to this equation.
Answer: no solution.
2. To determine the number of solutions for the equation 4x + 19 = -9 - 6x, we can simplify it:
Combining like terms, we get:
10x + 19 = -9
Subtracting 19 from both sides, we have:
10x = -28
Dividing both sides by 10, we find:
x = -2.8
This represents a single solution.
Answer: one solution.
3. To determine the number of solutions for the equation -2(6x - 4) = 8 - 12x, we can simplify it:
Expanding the left side, we get:
-12x + 8 = 8 - 12x
Notice that both -12x terms cancel out, leaving us with:
8 = 8
Since this equation is true, it means that any value of x will satisfy it.
Therefore, there are infinitely many solutions.
Answer: an infinite number of solutions.
4. The equations with only one solution are:
- 4(2x + 2) = 8x + 9. (After simplification, x = -5)
- 3x + (-4) = 6x - 4. (After simplification, x = -0.5)
- 9x - 9 = 3(3x - 3). (After simplification, x = -1)
Answer: 4(2x + 2) = 8x + 9, 3x + (-4) = 6x - 4, and 9x - 9 = 3(3x - 3).
5. The equations with no solution are:
- 7 + 12 = 8x + 29 - x, (After simplification, this becomes 5 = 7x + 29)
- 4 + 5x = 2(3x - 4), (After simplification, this becomes 5x - 4 = 6x - 8)
- 2(3x + 5) = 6x - 19, (After simplification, this becomes -6x - 10 = 6x - 19)
Answer: 7x + 12 = 8x + 29 - x, -4 + 5x = 2(3x - 4), -2(3x + 5) = 6x - 19.
To determine the number of solutions for each equation, we need to solve them and see the result. Here's how to do that:
1. Solve the equation 4x + 3 = 2(2x + 9):
- Simplify both sides: 4x + 3 = 4x + 18
- Subtract 4x from both sides: 3 = 18
- This is false, so there are no solutions. Answer: no solution.
2. Solve the equation 4x + 19 = -9 - 6x:
- Simplify both sides: 10x + 19 = -9
- Subtract 19 from both sides: 10x = -28
- Divide both sides by 10: x = -2.8
- There is only one solution. Answer: one solution.
3. Solve the equation -2(6x - 4) = 8 - 12x:
- Distribute -2 to (6x - 4): -12x + 8 = 8 - 12x
- Subtract -12x from both sides: 8 = 8
- This is true, meaning the equation is always true regardless of the value of x. Therefore, there are an infinite number of solutions. Answer: an infinite number of solutions.
Now let's go through the multiple-choice questions:
4. To identify equations with only one solution, we need to solve each equation mentioned:
- 4(2x + 2) = 8x + 9: Simplifying yields 8x + 8 = 8x + 9. Subtracting 8x from both sides gives 8 = 9. This is false, so there is no solution.
- 3x + (-4) = 6x - 4: Simplifying yields 3x - 4 = 6x - 4. Subtracting 3x from both sides gives -4 = 3x - 4. Adding 4 to both sides gives 0 = 3x. Dividing both sides by 3 gives x = 0. Thus, there is only one solution.
- 9x - 9 = 3(3x - 3): Simplifying yields 9x - 9 = 9x - 9. Subtracting 9x from both sides gives -9 = -9. This is true, so the equation is always true regardless of x. Therefore, there are an infinite number of solutions.
- 4x + 5x + x = 11x - 3: Simplifying yields 10x = 11x - 3. Subtracting 11x from both sides gives -x = -3. Dividing by -1 (which flips the inequality) gives x = 3. This implies that there is only one solution.
Answer: The equations that have only one solution are 3x + (-4) = 6x - 4 and 4x + 5x + x = 11x - 3.
5. To identify equations with no solution, we need to solve each equation mentioned:
- 7x - 12 = 8x + 29 - x: Simplifying yields 6x - 12 = 29. Adding 12 to both sides gives 6x = 41. Dividing both sides by 6 gives x = 41/6. Since x is not a real number, this implies no solution.
- (-4) + 5x = 2(3x - 4): Simplifying yields -4 + 5x = 6x - 8. Subtracting 6x from both sides gives -4 - x = -8. Adding x to both sides gives -4 = -8 + x. This simplifies to -4 = x - 8, which can be rewritten as -4 = x + (-8). This means that the equation is stating -4 is equal to some value of x that is 8 less than -8. Since this is impossible, no solution exists.
- 2(3x + 5) = 6x - 19: Simplifying yields 6x + 10 = 6x - 19. Subtracting 6x from both sides gives 10 = -19. This is false, so there is no solution.
- (-7) + 3x + 8x = 2(6x - 4): Simplifying yields -7 + 11x = 12x - 8. Adding 7 to both sides gives 11x = 12x - 1. Subtracting 12x from both sides gives -x = -1. Multiplying both sides by -1 (which flips the inequality) gives x = 1. However, when we substitute x = 1 back into the original equation, it does not hold true. Hence, no solution exists.
Answer: The equations that have no solution are 7x - 12 = 8x + 29 - x, -4 + 5x = 2(3x - 4), and -7 + 3x + 8x = 2(6x - 4).