Solve 3x+5/2-1-(x+2/3+x)<0

$$\frac { 3 x + 5 } { 2 } - 1 - ( \frac { x + 2 } { 3 } + x ) < 0$$

Solve for x

$$x<-5$$

Solution Steps

Multiply both sides of the equation by 6, the least common multiple of 2,3. Since 6 is positive, the inequality direction remains the same.

$$6\left(\frac{3x+5}{2}-1\right)-6\left(\frac{x+2}{3}+x\right)<0$$

Use the distributive property to multiply 6 by \frac{3x+5}{2}-1.

$$6\times \frac{3x+5}{2}-6-6\left(\frac{x+2}{3}+x\right)<0$$

Cancel out 2, the greatest common factor in 6 and 2.

$$3\left(3x+5\right)-6-6\left(\frac{x+2}{3}+x\right)<0$$

Use the distributive property to multiply 3 by 3x+5.

$$9x+15-6-6\left(\frac{x+2}{3}+x\right)<0$$

Subtract 6 from 15 to get 9.

$$9x+9-6\left(\frac{x+2}{3}+x\right)<0$$

Divide each term of x+2 by 3 to get \frac{1}{3}x+\frac{2}{3}.

$$9x+9-6\left(\frac{1}{3}x+\frac{2}{3}+x\right)<0$$

Combine \frac{1}{3}x and x to get \frac{4}{3}x.

$$9x+9-6\left(\frac{4}{3}x+\frac{2}{3}\right)<0$$

Use the distributive property to multiply -6 by \frac{4}{3}x+\frac{2}{3}.

$$9x+9-6\times \frac{4}{3}x-6\times \frac{2}{3}<0$$

Express -6\times \frac{4}{3} as a single fraction.

$$9x+9+\frac{-6\times 4}{3}x-6\times \frac{2}{3}<0$$

Multiply -6 and 4 to get -24.

$$9x+9+\frac{-24}{3}x-6\times \frac{2}{3}<0$$

Divide -24 by 3 to get -8.

$$9x+9-8x-6\times \frac{2}{3}<0$$

Express -6\times \frac{2}{3} as a single fraction.

$$9x+9-8x+\frac{-6\times 2}{3}<0$$

Multiply -6 and 2 to get -12.

$$9x+9-8x+\frac{-12}{3}<0$$

Divide -12 by 3 to get -4.

$$9x+9-8x-4<0$$

Combine 9x and -8x to get x.

$$x+9-4<0$$

Subtract 4 from 9 to get 5.

$$x+5<0$$

Subtract 5 from both sides. Anything subtracted from zero gives its negation.

$$x<-5$$