Solve 5x^2+7x/4-2x^2-9x/2leq0

$$\frac { 5 x ^ { 2 } + 7 x } { 4 } - \frac { 2 x ^ { 2 } - 9 x } { 2 } \leq 0$$

Solve for x

$$x\in \begin{bmatrix}-25,0\end{bmatrix}$$

Solution Steps

Multiply both sides of the equation by 4, the least common multiple of 4,2. Since 4 is positive, the inequality direction remains the same.

$$5x^{2}+7x-2\left(2x^{2}-9x\right)\leq 0$$

Use the distributive property to multiply -2 by 2x^{2}-9x.

$$5x^{2}+7x-4x^{2}+18x\leq 0$$

Combine 5x^{2} and -4x^{2} to get x^{2}.

$$x^{2}+7x+18x\leq 0$$

Combine 7x and 18x to get 25x.

$$x^{2}+25x\leq 0$$

Factor out x.

$$x\left(x+25\right)\leq 0$$

For the product to be ≤0, one of the values x+25 and x has to be ≥0 and the other has to be ≤0. Consider the case when x+25\geq 0 and x\leq 0.

$$x+25\geq 0 x\leq 0$$

The solution satisfying both inequalities is x\in \left[-25,0\right].

$$x\in \begin{bmatrix}-25,0\end{bmatrix}$$

Consider the case when x+25\leq 0 and x\geq 0.

$$x\geq 0 x+25\leq 0$$

This is false for any x.

$$x\in \emptyset$$

The final solution is the union of the obtained solutions.

$$x\in \begin{bmatrix}-25,0\end{bmatrix}$$