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

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

Solve for x

$$x<-7$$

Solution Steps

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

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

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

$$x\times \frac{1}{2}+\frac{1}{2}<2x\times \frac{1}{5}-\frac{1}{5}$$

Multiply 2 and \frac{1}{5} to get \frac{2}{5}.

$$x\times \frac{1}{2}+\frac{1}{2}<\frac{2}{5}x-\frac{1}{5}$$

Subtract \frac{2}{5}x from both sides.

$$x\times \frac{1}{2}+\frac{1}{2}-\frac{2}{5}x<-\frac{1}{5}$$

Combine x\times \frac{1}{2} and -\frac{2}{5}x to get \frac{1}{10}x.

$$\frac{1}{10}x+\frac{1}{2}<-\frac{1}{5}$$

Subtract \frac{1}{2} from both sides.

$$\frac{1}{10}x<-\frac{1}{5}-\frac{1}{2}$$

Least common multiple of 5 and 2 is 10. Convert -\frac{1}{5} and \frac{1}{2} to fractions with denominator 10.

$$\frac{1}{10}x<-\frac{2}{10}-\frac{5}{10}$$

Since -\frac{2}{10} and \frac{5}{10} have the same denominator, subtract them by subtracting their numerators.

$$\frac{1}{10}x<\frac{-2-5}{10}$$

Subtract 5 from -2 to get -7.

$$\frac{1}{10}x<-\frac{7}{10}$$

Multiply both sides by 10, the reciprocal of \frac{1}{10}. Since \frac{1}{10} is positive, the inequality direction remains the same.

$$x<-\frac{7}{10}\times 10$$

Cancel out 10 and 10.

$$x<-7$$