Solve 5-2x/3+4<3x-5/4*3x/2

$$\frac { 5 - 2 x } { 3 } + 4 < \frac { 3 x - 5 } { 4 } \cdot \frac { 3 x } { 2 }$$

Solve for x

$$x\in \left(-\infty,\frac{29-\sqrt{15529}}{54}\right)\cup \left(\frac{\sqrt{15529}+29}{54},\infty\right)$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 12, the least common multiple of 3,4,2. Since 12 is positive, the inequality direction remains the same.

$$4\left(5-2x\right)+48<3\left(3x-5\right)\times \frac{3x}{2}$$

Use the distributive property to multiply 4 by 5-2x.

$$20-8x+48<3\left(3x-5\right)\times \frac{3x}{2}$$

Add 20 and 48 to get 68.

$$68-8x<3\left(3x-5\right)\times \frac{3x}{2}$$

Express 3\times \frac{3x}{2} as a single fraction.

$$68-8x<\frac{3\times 3x}{2}\left(3x-5\right)$$

Use the distributive property to multiply \frac{3\times 3x}{2} by 3x-5.

$$68-8x<3\times \frac{x\times 3^{2}}{2}x-5\times \frac{3\times 3x}{2}$$

Calculate 3 to the power of 2 and get 9.

$$68-8x<3\times \frac{x\times 9}{2}x-5\times \frac{3\times 3x}{2}$$

Express 3\times \frac{x\times 9}{2} as a single fraction.

$$68-8x<\frac{3x\times 9}{2}x-5\times \frac{3\times 3x}{2}$$

Express \frac{3x\times 9}{2}x as a single fraction.

$$68-8x<\frac{3x\times 9x}{2}-5\times \frac{3\times 3x}{2}$$

Multiply 3 and 3 to get 9.

$$68-8x<\frac{3x\times 9x}{2}-5\times \frac{9x}{2}$$

Express -5\times \frac{9x}{2} as a single fraction.

$$68-8x<\frac{3x\times 9x}{2}+\frac{-5\times 9x}{2}$$

Since \frac{3x\times 9x}{2} and \frac{-5\times 9x}{2} have the same denominator, add them by adding their numerators.

$$68-8x<\frac{3x\times 9x-5\times 9x}{2}$$

Do the multiplications in 3x\times 9x-5\times 9x.

$$68-8x<\frac{27x^{2}-45x}{2}$$

Divide each term of 27x^{2}-45x by 2 to get \frac{27}{2}x^{2}-\frac{45}{2}x.

$$68-8x<\frac{27}{2}x^{2}-\frac{45}{2}x$$

Subtract \frac{27}{2}x^{2} from both sides.

$$68-8x-\frac{27}{2}x^{2}<-\frac{45}{2}x$$

Add \frac{45}{2}x to both sides.

$$68-8x-\frac{27}{2}x^{2}+\frac{45}{2}x<0$$

Combine -8x and \frac{45}{2}x to get \frac{29}{2}x.

$$68+\frac{29}{2}x-\frac{27}{2}x^{2}<0$$

Multiply the inequality by -1 to make the coefficient of the highest power in 68+\frac{29}{2}x-\frac{27}{2}x^{2} positive. Since -1 is negative, the inequality direction is changed.

$$-68-\frac{29}{2}x+\frac{27}{2}x^{2}>0$$

To solve the inequality, factor the left hand side. Quadratic polynomial can be factored using the transformation ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right), where x_{1} and x_{2} are the solutions of the quadratic equation ax^{2}+bx+c=0.

$$-68-\frac{29}{2}x+\frac{27}{2}x^{2}=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. Substitute \frac{27}{2} for a, -\frac{29}{2} for b, and -68 for c in the quadratic formula.

$$x=\frac{-\left(-\frac{29}{2}\right)±\sqrt{\left(-\frac{29}{2}\right)^{2}-4\times \frac{27}{2}\left(-68\right)}}{2\times \frac{27}{2}}$$

Do the calculations.

$$x=\frac{\frac{29}{2}±\frac{1}{2}\sqrt{15529}}{27}$$

Solve the equation x=\frac{\frac{29}{2}±\frac{1}{2}\sqrt{15529}}{27} when ± is plus and when ± is minus.

$$x=\frac{\sqrt{15529}+29}{54} x=\frac{29-\sqrt{15529}}{54}$$

Rewrite the inequality by using the obtained solutions.

$$\frac{27}{2}\left(x-\frac{\sqrt{15529}+29}{54}\right)\left(x-\frac{29-\sqrt{15529}}{54}\right)>0$$

For the product to be positive, x-\frac{\sqrt{15529}+29}{54} and x-\frac{29-\sqrt{15529}}{54} have to be both negative or both positive. Consider the case when x-\frac{\sqrt{15529}+29}{54} and x-\frac{29-\sqrt{15529}}{54} are both negative.

$$x-\frac{\sqrt{15529}+29}{54}<0 x-\frac{29-\sqrt{15529}}{54}<0$$

The solution satisfying both inequalities is x<\frac{29-\sqrt{15529}}{54}.

$$x<\frac{29-\sqrt{15529}}{54}$$

Consider the case when x-\frac{\sqrt{15529}+29}{54} and x-\frac{29-\sqrt{15529}}{54} are both positive.

$$x-\frac{29-\sqrt{15529}}{54}>0 x-\frac{\sqrt{15529}+29}{54}>0$$

The solution satisfying both inequalities is x>\frac{\sqrt{15529}+29}{54}.

$$x>\frac{\sqrt{15529}+29}{54}$$

The final solution is the union of the obtained solutions.

$$x<\frac{29-\sqrt{15529}}{54}\text{; }x>\frac{\sqrt{15529}+29}{54}$$