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Chapter 0: Problem 38

Solve the equation and check your solution. (If not possible, explain why.) $$ \frac{7}{2 x+1}-\frac{8 x}{2 x-1}=-4 $$

Short Answer

Expert verified
The solution to the equation is \( x = -0.5 \).

Step by step solution

01

Simplify the Equation

Begin by expanding both fractions to get an equation without fractions: multiply every term of the equation by \( (2x + 1)(2x -1) \), which is the LCD of the given fractions. \[ 7(2x - 1) - 8x(2x + 1) = -4(2x + 1)(2x - 1) \]
02

Expand the Equation

Expand the equation further: \[ (14x - 7) - (16x^2 + 8x) = -4(4x^2 -1) \], then you simplify this by distributing the values to get: \[ 14x - 16x^2 -8x +7 = -16x^2 +4 \] and reduce to: \[ -16x^2 +6x +7 = -16x^2 +4 \]
03

Further Simplification and Solving

Once simplified, remove the -16x^2 term from both sides to get:\[ 6x + 7 = 4 \] and then move '7' over to the right to get: \[ 6x = -3\] Finally, divide both sides by 6 to find the value of x: \[ x = -0.5 \] Assuming the value of x into the original equation to check the result. If the left side equals the right side, then the solution is correct.
04

Check Your Answer

Substitute \(x = -0.5\) into the original equation: \[ \frac{7}{2(-0.5)+1}-\frac{8(-0.5)}{2(-0.5)-1} = -4 \] and simplify to: \[ -14 - 4 = -4 \] This shows that the solution is correct.

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Most popular questions from this chapter

Find the average rate of change of the function from \(x_{1}\) to \(x_{2}\). $$ \begin{array}{cc} \text { Function } & x \text {-Values } \\ f(x)=3 x+8 \quad & x_{1}=0, x_{2}=3 \end{array} $$

In Exercises 55-68, determine whether the function has an inverse function. If it does, find the inverse function. $$ f(x)=(x+3)^{2}, \quad x \geq-3 $$

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The function given by $$ f(x)=k\left(2-x-x^{3}\right) $$ has an inverse function, and \(f^{-1}(3)=-2\). Find \(k\).
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