Solve (x ^ 2)/(x + 4) > x - 1 | Equatix - Math Solver

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Example

3(r+2s)=2t-4

a⋅(b-2)=3b

3(r+2s)=2t-4

a⋅(b-2)=3b

Question

$$\frac{x^{2}}{x+4}>x-1$$

Solve for x

$x\in \left(-4,\frac{4}{3}\right)$

Solution Steps

Subtract $x$ from both sides.

$$\frac{x^{2}}{x+4}-x>-1$$

To add or subtract expressions, expand them to make their denominators the same. Multiply $x$ times $\frac{x+4}{x+4}$.

$$\frac{x^{2}}{x+4}-\frac{x\left(x+4\right)}{x+4}>-1$$

Since $\frac{x^{2}}{x+4}$ and $\frac{x\left(x+4\right)}{x+4}$ have the same denominator, subtract them by subtracting their numerators.

$$\frac{x^{2}-x\left(x+4\right)}{x+4}>-1$$

Do the multiplications in $x^{2}-x\left(x+4\right)$.

$$\frac{x^{2}-x^{2}-4x}{x+4}>-1$$

Combine like terms in $x^{2}-x^{2}-4x$.

$$\frac{-4x}{x+4}>-1$$

Denominator $x+4$ cannot be zero since division by zero is not defined. There are two cases.

$$x+4>0$$ $$x+4<0$$

Consider the case when $x+4$ is positive. Move $4$ to the right hand side.

$$x>-4$$

The initial inequality does not change the direction when multiplied by $x+4$ for $x+4>0$.

$$-4x>-\left(x+4\right)$$

Multiply out the right hand side.

$$-4x>-x-4$$

Move the terms containing $x$ to the left hand side and all other terms to the right hand side.

$$-4x+x>-4$$

Combine like terms.

$$-3x>-4$$

Divide both sides by $-3$. Since $-3$ is negative, the inequality direction is changed.

$$x<\frac{4}{3}$$

Consider condition $x>-4$ specified above.

$$x\in \left(-4,\frac{4}{3}\right)$$

Now consider the case when $x+4$ is negative. Move $4$ to the right hand side.

$$x<-4$$

The initial inequality changes the direction when multiplied by $x+4$ for $x+4<0$.

$$-4x<-\left(x+4\right)$$

Multiply out the right hand side.

$$-4x<-x-4$$

Move the terms containing $x$ to the left hand side and all other terms to the right hand side.

$$-4x+x<-4$$

Combine like terms.

$$-3x<-4$$

Divide both sides by $-3$. Since $-3$ is negative, the inequality direction is changed.

$$x>\frac{4}{3}$$

Consider condition $x<-4$ specified above.

$$x\in \emptyset$$

The final solution is the union of the obtained solutions.

$$x\in \left(-4,\frac{4}{3}\right)$$