To determine which equation has the solution set \( x = {-\frac{9}{5}, 3} \), we need to check each option one by one to see which one produces these roots.
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Equation: \((x-3)(9x+5)=0\)
To find the solutions, set each factor to zero:
- \(x - 3 = 0 \implies x = 3\)
- \(9x + 5 = 0 \implies 9x = -5 \implies x = -\frac{5}{9}\)
This gives us the solution set \( x = {3, -\frac{5}{9}} \), which does not match.
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Equation: \((x-3)(5x+9)=0\)
Set each factor to zero:
- \(x - 3 = 0 \implies x = 3\)
- \(5x + 9 = 0 \implies 5x = -9 \implies x = -\frac{9}{5}\)
This gives us the solution set \( x = {3, -\frac{9}{5}} \), which matches.
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Equation: \((x+3)(9x-5)=0\)
Set each factor to zero:
- \(x + 3 = 0 \implies x = -3\)
- \(9x - 5 = 0 \implies 9x = 5 \implies x = \frac{5}{9}\)
This gives us the solution set \( x = {-3, \frac{5}{9}} \), which does not match.
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Equation: \(-3x(5x+9)=0\)
Set each factor to zero:
- \(-3x = 0 \implies x = 0\)
- \(5x + 9 = 0 \implies 5x = -9 \implies x = -\frac{9}{5}\)
This gives us the solution set \( x = {0, -\frac{9}{5}} \), which does not match.
Thus, the equation that has the solution set \( x = {-\frac{9}{5}, 3} \) is:
(x−3)(5x+9)=0.