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Sagot :
To solve the equation [tex]\( x - 2(x + 3) = 4(2x + 3) - (x + 4) \)[/tex], we need to simplify both sides of the equation step by step and find equivalent equations that Katrina might use. Here is the detailed step-by-step solution:
1. Simplify the Left Side:
[tex]\[ x - 2(x + 3) \][/tex]
Distribute the [tex]\(-2\)[/tex]:
[tex]\[ x - 2x - 6 \][/tex]
Combine like terms:
[tex]\[ -x - 6 \][/tex]
2. Simplify the Right Side:
[tex]\[ 4(2x + 3) - (x + 4) \][/tex]
Distribute the [tex]\(4\)[/tex]:
[tex]\[ 8x + 12 - (x + 4) \][/tex]
Distribute the [tex]\(-1\)[/tex] in front of the [tex]\((x + 4)\)[/tex]:
[tex]\[ 8x + 12 - x - 4 \][/tex]
Combine like terms:
[tex]\[ 7x + 8 \][/tex]
3. Set the simplified sides equal to each other:
[tex]\[ -x - 6 = 7x + 8 \][/tex]
So, Katrina might use the equation:
[tex]\[ -x - 6 = 7x + 8 \][/tex]
Now, let's expand on which of the original options match this or could be intermediate steps:
1. Option [tex]\[ x - 2x - 6 = 8x + 12 - x - 4 \][/tex]
This simplification step fits correctly:
[tex]\[ x - 2x - 6 = -x - 6\][/tex]
[tex]\[ 8x + 12 - x - 4 = 7x + 8 \][/tex]
2. Option [tex]\[ x - 2x + 6 = 8x + 12 - x + 4 \][/tex]
This option does not work because the [tex]\(\ 6\)[/tex] should be [tex]\(-6\)[/tex] on the left.
3. Option [tex]\[ x - 2x - 6 = 8x + 12 - x + 4 \][/tex]
This option does not work because the [tex]\( 4\)[/tex] should be [tex]\(-4\)[/tex] on the right.
4. Option [tex]\[ x - 2x + 3 = 8x + 3 - x + 4 \][/tex]
This option does not work.
5. Option [tex]\[ -x - 6 = 7x + 8 \][/tex]
This is an equivalent equation as derived above.
6. Option [tex]\[ -x + 6 = 7x + 16 \][/tex]
This option does not work, since the numbers on the left and right do not match with our simplifications.
7. Option [tex]\[ -x - 6 = 7x + 16 \][/tex]
This option does not work, since the numbers on the right do not match our simplifications.
8. Option [tex]\[ -x + 3 - 7x + 7 \][/tex]
This option does not work as it does not represent an equivalent form of the equation correctly.
Therefore, the following equivalent equations match:
- [tex]\( x - 2x - 6 = 8x + 12 - x - 4 \)[/tex]
- [tex]\( -x - 6 = 7x + 8 \)[/tex]
1. Simplify the Left Side:
[tex]\[ x - 2(x + 3) \][/tex]
Distribute the [tex]\(-2\)[/tex]:
[tex]\[ x - 2x - 6 \][/tex]
Combine like terms:
[tex]\[ -x - 6 \][/tex]
2. Simplify the Right Side:
[tex]\[ 4(2x + 3) - (x + 4) \][/tex]
Distribute the [tex]\(4\)[/tex]:
[tex]\[ 8x + 12 - (x + 4) \][/tex]
Distribute the [tex]\(-1\)[/tex] in front of the [tex]\((x + 4)\)[/tex]:
[tex]\[ 8x + 12 - x - 4 \][/tex]
Combine like terms:
[tex]\[ 7x + 8 \][/tex]
3. Set the simplified sides equal to each other:
[tex]\[ -x - 6 = 7x + 8 \][/tex]
So, Katrina might use the equation:
[tex]\[ -x - 6 = 7x + 8 \][/tex]
Now, let's expand on which of the original options match this or could be intermediate steps:
1. Option [tex]\[ x - 2x - 6 = 8x + 12 - x - 4 \][/tex]
This simplification step fits correctly:
[tex]\[ x - 2x - 6 = -x - 6\][/tex]
[tex]\[ 8x + 12 - x - 4 = 7x + 8 \][/tex]
2. Option [tex]\[ x - 2x + 6 = 8x + 12 - x + 4 \][/tex]
This option does not work because the [tex]\(\ 6\)[/tex] should be [tex]\(-6\)[/tex] on the left.
3. Option [tex]\[ x - 2x - 6 = 8x + 12 - x + 4 \][/tex]
This option does not work because the [tex]\( 4\)[/tex] should be [tex]\(-4\)[/tex] on the right.
4. Option [tex]\[ x - 2x + 3 = 8x + 3 - x + 4 \][/tex]
This option does not work.
5. Option [tex]\[ -x - 6 = 7x + 8 \][/tex]
This is an equivalent equation as derived above.
6. Option [tex]\[ -x + 6 = 7x + 16 \][/tex]
This option does not work, since the numbers on the left and right do not match with our simplifications.
7. Option [tex]\[ -x - 6 = 7x + 16 \][/tex]
This option does not work, since the numbers on the right do not match our simplifications.
8. Option [tex]\[ -x + 3 - 7x + 7 \][/tex]
This option does not work as it does not represent an equivalent form of the equation correctly.
Therefore, the following equivalent equations match:
- [tex]\( x - 2x - 6 = 8x + 12 - x - 4 \)[/tex]
- [tex]\( -x - 6 = 7x + 8 \)[/tex]
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