Solve 1/2x+2-1/5x-3/5<-1/10x

$$\frac { 1 } { 2 } x + 2 - \frac { 1 } { 5 } x - \frac { 3 } { 5 } < - \frac { 1 } { 10 } x$$

Solve for x

$$x<-\frac{7}{2}$$

Solution Steps

Combine \frac{1}{2}x and -\frac{1}{5}x to get \frac{3}{10}x.

$$\frac{3}{10}x+2-\frac{3}{5}<-\frac{1}{10}x$$

Convert 2 to fraction \frac{10}{5}.

$$\frac{3}{10}x+\frac{10}{5}-\frac{3}{5}<-\frac{1}{10}x$$

Since \frac{10}{5} and \frac{3}{5} have the same denominator, subtract them by subtracting their numerators.

$$\frac{3}{10}x+\frac{10-3}{5}<-\frac{1}{10}x$$

Subtract 3 from 10 to get 7.

$$\frac{3}{10}x+\frac{7}{5}<-\frac{1}{10}x$$

Add \frac{1}{10}x to both sides.

$$\frac{3}{10}x+\frac{7}{5}+\frac{1}{10}x<0$$

Combine \frac{3}{10}x and \frac{1}{10}x to get \frac{2}{5}x.

$$\frac{2}{5}x+\frac{7}{5}<0$$

Subtract \frac{7}{5} from both sides. Anything subtracted from zero gives its negation.

$$\frac{2}{5}x<-\frac{7}{5}$$

Multiply both sides by \frac{5}{2}, the reciprocal of \frac{2}{5}. Since \frac{2}{5} is positive, the inequality direction remains the same.

$$x<-\frac{7}{5}\times \frac{5}{2}$$

Multiply -\frac{7}{5} times \frac{5}{2} by multiplying numerator times numerator and denominator times denominator.

$$x<\frac{-7\times 5}{5\times 2}$$

Cancel out 5 in both numerator and denominator.

$$x<\frac{-7}{2}$$

Fraction \frac{-7}{2} can be rewritten as -\frac{7}{2} by extracting the negative sign.

$$x<-\frac{7}{2}$$