What value of [tex]$x$[/tex] solves the equation?
[tex]\frac{3 x+9}{2}=\frac{x-4}{3}[/tex]
Answer (2)
The solution to the equation 2 3 x + 9 = 3 x − 4 is found by clearing fractions, expanding, and isolating x, leading to the final answer of x = -5.
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Multiply both sides by 6 to eliminate fractions: 3 ( 3 x + 9 ) = 2 ( x − 4 ) .
Expand both sides: 9 x + 27 = 2 x − 8 .
Simplify and isolate x : 7 x = − 35 .
Solve for x : x = − 5 . The solution to the equation is − 5 .
Explanation
Understanding the Problem We are given the equation 2 3 x + 9 = 3 x − 4 and we need to find the value of x that satisfies this equation.
Eliminating Fractions To solve the equation, we first eliminate the fractions by multiplying both sides of the equation by the least common multiple (LCM) of the denominators, which is 6. This gives us: 6 ⋅ 2 3 x + 9 = 6 ⋅ 3 x − 4
Simplifying the Equation Now, simplify both sides of the equation: 3 ( 3 x + 9 ) = 2 ( x − 4 )
Expanding the Equation Next, expand both sides of the equation by distributing the constants: 9 x + 27 = 2 x − 8
Isolating x Now, we want to isolate x on one side of the equation. Subtract 2 x from both sides: 9 x − 2 x + 27 = 2 x − 2 x − 8 7 x + 27 = − 8
Further Isolating x Subtract 27 from both sides: 7 x + 27 − 27 = − 8 − 27 7 x = − 35
Solving for x Finally, divide both sides by 7 to solve for x :
7 7 x = 7 − 35 x = − 5
Final Answer Thus, the value of x that solves the equation is -5.
Examples
Consider a situation where you are comparing two different investment options. Option A gives you a return of 3 x + 9 dollars for every 2 units of investment, while Option B gives you a return of x − 4 dollars for every 3 units of investment. Solving the equation 2 3 x + 9 = 3 x − 4 helps you find the value of x for which the returns from both investment options are equal. This type of problem is a linear equation, and solving it allows you to make informed decisions in various financial and comparative scenarios.
