Solved on Jan 22, 2024

1. Match function rules to invalid inputs: A. 3/x, B. (x+4)/3, C. 1/(x-3). Invalid inputs: 1. 0, 2. -4, 3. 1.
2. Find $x$ that satisfies $-(-2x+1)=9-14x$ . Explain and check the solution.

STEP 1

Assumptions for problem 3
1. We are given three function rules: A, B, and C.
2. We need to match each function rule with a value that could not be a possible input for that function.
3. The values to consider as possible inputs are 1.3, 4, -4, 0, and 1.

STEP 2

Identify the inputs that would cause a problem for each function rule.
For function A, the rule is "3 divided by the input". We cannot divide by zero, so the input that cannot be used for function A is 0.
For function B, the rule is "Add 4 to the input, then divide this value into 3". There is no restriction from this rule that would prevent any of the given numbers from being used as an input.
For function C, the rule is "Subtract 3 from the input, then divide this value into 1". We cannot divide by zero, so the input that cannot be used for function C is 3.

STEP 3

Match the function rules with the values that cannot be possible inputs.
A. 3 divided by the input cannot have the input 0, so the match is A-4.
B. Add 4 to the input, then divide this value into 3 does not have a restriction, so it could potentially accept all given inputs. However, since we need to match the functions with the given values, and we have already matched 0 with A, we need to find the value that is left unmatched for B. Since 1.3, -4, and 1 do not cause any issues with the other functions, the remaining value, 4, must be matched with B by process of elimination.
C. Subtract 3 from the input, then divide this value into 1 cannot have the input 3, so the match is C-2.

STEP 4

The matches for problem 3 are:
A-4, B-2, C-3.
Now, we will move on to problem 4.

STEP 5

Assumptions for problem 4
1. We are given the equation $-(-2x+1)=9-14x$ .
2. We need to find a value of $x$ that makes the equation true.

STEP 6

Simplify the left side of the equation by distributing the negative sign.
$-(-2x+1) = -1 \cdot (-2x) + -1 \cdot 1$

STEP 7

Continue simplifying the left side.
$-(-2x+1) = 2x - 1$

STEP 8

Now we have a simplified equation:
$2x - 1 = 9 - 14x$

STEP 9

Add $14x$ to both sides of the equation to get all the $x$ terms on one side.
$2x + 14x - 1 = 9 - 14x + 14x$

STEP 10

Combine like terms.
$16x - 1 = 9$

STEP 11

Add 1 to both sides to isolate the term with $x$ .
$16x - 1 + 1 = 9 + 1$

STEP 12

Simplify both sides of the equation.
$16x = 10$

STEP 13

Divide both sides by 16 to solve for $x$ .
$\frac{16x}{16} = \frac{10}{16}$

STEP 14

Simplify the fraction on the right side.
$x = \frac{5}{8}$

STEP 15

Check the solution by substituting $x = \frac{5}{8}$ back into the original equation.
$-(-2 \cdot \frac{5}{8} + 1) = 9 - 14 \cdot \frac{5}{8}$

STEP 16

Perform the multiplication.
$-(-\frac{10}{8} + 1) = 9 - \frac{70}{8}$

STEP 17

Simplify the fractions by finding a common denominator.
$-(-\frac{10}{8} + \frac{8}{8}) = \frac{72}{8} - \frac{70}{8}$

STEP 18

Combine the terms inside the parentheses and simplify the right side.
$-(-\frac{2}{8}) = \frac{2}{8}$

STEP 19

Simplify the left side by distributing the negative sign.
$\frac{2}{8} = \frac{2}{8}$

STEP 20

Since the left side equals the right side, our solution $x = \frac{5}{8}$ is correct.
The value of $x$ that makes the equation true is $x = \frac{5}{8}$ .

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1. Match function rules to invalid inputs: A. 3/x, B. (x+4)/3, C. 1/(x-3). Invalid inputs: 1. 0, 2. -4, 3. 1.2. Find xxx that satisfies −(−2x+1)=9−14x-(-2x+1)=9-14x−(−2x+1)=9−14x. Explain and check the solution.

STEP 1

Assumptions for problem 31. We are given three function rules: A, B, and C.2. We need to match each function rule with a value that could not be a possible input for that function.3. The values to consider as possible inputs are 1.3, 4, -4, 0, and 1.

STEP 2

STEP 3

STEP 4

The matches for problem 3 are:A-4, B-2, C-3.Now, we will move on to problem 4.

STEP 5

Assumptions for problem 41. We are given the equation −(−2x+1)=9−14x-(-2x+1)=9-14x−(−2x+1)=9−14x.2. We need to find a value of xxx that makes the equation true.

STEP 6

Simplify the left side of the equation by distributing the negative sign.−(−2x+1)=−1⋅(−2x)+−1⋅1-(-2x+1) = -1 \cdot (-2x) + -1 \cdot 1−(−2x+1)=−1⋅(−2x)+−1⋅1

STEP 7

Continue simplifying the left side.−(−2x+1)=2x−1-(-2x+1) = 2x - 1−(−2x+1)=2x−1

STEP 8

Now we have a simplified equation:2x−1=9−14x2x - 1 = 9 - 14x2x−1=9−14x

STEP 9

Add 14x14x14x to both sides of the equation to get all the xxx terms on one side.2x+14x−1=9−14x+14x2x + 14x - 1 = 9 - 14x + 14x2x+14x−1=9−14x+14x

STEP 10

Combine like terms.16x−1=916x - 1 = 916x−1=9

STEP 11

Add 1 to both sides to isolate the term with xxx.16x−1+1=9+116x - 1 + 1 = 9 + 116x−1+1=9+1

STEP 12

Simplify both sides of the equation.16x=1016x = 1016x=10

STEP 13

Divide both sides by 16 to solve for xxx.16x16=1016\frac{16x}{16} = \frac{10}{16}1616x​=1610​

STEP 14

Simplify the fraction on the right side.x=58x = \frac{5}{8}x=85​

STEP 15

Check the solution by substituting x=58x = \frac{5}{8}x=85​ back into the original equation.−(−2⋅58+1)=9−14⋅58-(-2 \cdot \frac{5}{8} + 1) = 9 - 14 \cdot \frac{5}{8}−(−2⋅85​+1)=9−14⋅85​

STEP 16

Perform the multiplication.−(−108+1)=9−708-(-\frac{10}{8} + 1) = 9 - \frac{70}{8}−(−810​+1)=9−870​

STEP 17

Simplify the fractions by finding a common denominator.−(−108+88)=728−708-(-\frac{10}{8} + \frac{8}{8}) = \frac{72}{8} - \frac{70}{8}−(−810​+88​)=872​−870​

STEP 18

Combine the terms inside the parentheses and simplify the right side.−(−28)=28-(-\frac{2}{8}) = \frac{2}{8}−(−82​)=82​

STEP 19

Simplify the left side by distributing the negative sign.28=28\frac{2}{8} = \frac{2}{8}82​=82​

STEP 20

Since the left side equals the right side, our solution x=58x = \frac{5}{8}x=85​ is correct.The value of xxx that makes the equation true is x=58x = \frac{5}{8}x=85​.

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1. Match function rules to invalid inputs: A. 3/x, B. (x+4)/3, C. 1/(x-3). Invalid inputs: 1. 0, 2. -4, 3. 1.
2. Find $x$ that satisfies $-(-2x+1)=9-14x$ . Explain and check the solution.

Assumptions for problem 3
1. We are given three function rules: A, B, and C.
2. We need to match each function rule with a value that could not be a possible input for that function.
3. The values to consider as possible inputs are 1.3, 4, -4, 0, and 1.

The matches for problem 3 are:
A-4, B-2, C-3.
Now, we will move on to problem 4.

Assumptions for problem 4
1. We are given the equation $-(-2x+1)=9-14x$ .
2. We need to find a value of $x$ that makes the equation true.

Simplify the left side of the equation by distributing the negative sign.
$-(-2x+1) = -1 \cdot (-2x) + -1 \cdot 1$

Continue simplifying the left side.
$-(-2x+1) = 2x - 1$

Now we have a simplified equation:
$2x - 1 = 9 - 14x$

Add $14x$ to both sides of the equation to get all the $x$ terms on one side.
$2x + 14x - 1 = 9 - 14x + 14x$

Combine like terms.
$16x - 1 = 9$

Add 1 to both sides to isolate the term with $x$ .
$16x - 1 + 1 = 9 + 1$

Simplify both sides of the equation.
$16x = 10$

Divide both sides by 16 to solve for $x$ .
$\frac{16x}{16} = \frac{10}{16}$

Simplify the fraction on the right side.
$x = \frac{5}{8}$

Check the solution by substituting $x = \frac{5}{8}$ back into the original equation.
$-(-2 \cdot \frac{5}{8} + 1) = 9 - 14 \cdot \frac{5}{8}$

Perform the multiplication.
$-(-\frac{10}{8} + 1) = 9 - \frac{70}{8}$

Simplify the fractions by finding a common denominator.
$-(-\frac{10}{8} + \frac{8}{8}) = \frac{72}{8} - \frac{70}{8}$

Combine the terms inside the parentheses and simplify the right side.
$-(-\frac{2}{8}) = \frac{2}{8}$

Simplify the left side by distributing the negative sign.
$\frac{2}{8} = \frac{2}{8}$

Since the left side equals the right side, our solution $x = \frac{5}{8}$ is correct.
The value of $x$ that makes the equation true is $x = \frac{5}{8}$ .