Algebra 1 A Semester Exam17 of 4017 of 40 ItemsQuestionWhich of the following equations has the solution set x={−9/5,3}?(1 point)Responses(x+3)(9x−5)=0left parenthesis x plus 3 right parenthesis left parenthesis 9 x minus 5 right parenthesis equals 0(x−3)(9x+5)=0left parenthesis x minus 3 right parenthesis left parenthesis 9 x plus 5 right parenthesis equals 0(x−3)(5x+9)=0left parenthesis x minus 3 right parenthesis left parenthesis 5 x plus 9 right parenthesis equals 0−3x(5x+9)=0

Answer 1

To find the equation that has the solution set \( x = {-\frac{9}{5}, 3} \), we can set up factors based on the solutions.

The solutions \( x = 3 \) and \( x = -\frac{9}{5} \) can be represented as factors of an equation:

1. For \( x = 3 \), the factor is \( (x - 3) \).
2. For \( x = -\frac{9}{5} \), the factor is \( \left(x + \frac{9}{5}\right) \).

To combine these into a single equation, we can write:

\[ (x - 3)\left(x + \frac{9}{5}\right) = 0 \]

To eliminate the fraction, we can multiply both sides by 5:

\[ 5(x - 3)\left(x + \frac{9}{5}\right) = 0 \] This becomes:

\[ 5(x - 3)(5x + 9) = 0 \]

Now, let's verify that this expands correctly and determines if we have found the right equation by checking the answers provided:

1. Option (x + 3)(9x - 5) = 0: This gives solutions \( x = -3 \) and \( x = \frac{5}{9} \). Not correct.
2. Option (x - 3)(9x + 5) = 0: This gives solutions \( x = 3 \) and \( x = -\frac{5}{9} \). Not correct.
3. Option (x - 3)(5x + 9) = 0: This gives \( x = 3 \) and \( 5x + 9 = 0 \Rightarrow x = -\frac{9}{5} \). This is correct.
4. Option -3x(5x + 9) = 0: This gives solution \( 0 \) from \( -3x = 0 \), and no valid solutions from \( 5x + 9 = 0 \). Not correct.

The correct option is:

(x - 3)(5x + 9) = 0.