Solve 0.1x+0.5/0.2-1<0.3x+0.2/0.3

$$\frac{ 0.1x+0.5 }{ 0.2 } -1 < \frac{ 0.3x+0.2 }{ 0.3 }$$

Solve for x

$$x>\frac{5}{3}$$

Solution Steps

Divide each term of 0.1x+0.5 by 0.2 to get \frac{0.1x}{0.2}+\frac{0.5}{0.2}.

$$\frac{0.1x}{0.2}+\frac{0.5}{0.2}-1<\frac{0.3x+0.2}{0.3}$$

Divide 0.1x by 0.2 to get 0.5x.

$$0.5x+\frac{0.5}{0.2}-1<\frac{0.3x+0.2}{0.3}$$

Expand \frac{0.5}{0.2} by multiplying both numerator and the denominator by 10.

$$0.5x+\frac{5}{2}-1<\frac{0.3x+0.2}{0.3}$$

Convert 1 to fraction \frac{2}{2}.

$$0.5x+\frac{5}{2}-\frac{2}{2}<\frac{0.3x+0.2}{0.3}$$

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

$$0.5x+\frac{5-2}{2}<\frac{0.3x+0.2}{0.3}$$

Subtract 2 from 5 to get 3.

$$0.5x+\frac{3}{2}<\frac{0.3x+0.2}{0.3}$$

Divide each term of 0.3x+0.2 by 0.3 to get \frac{0.3x}{0.3}+\frac{0.2}{0.3}.

$$0.5x+\frac{3}{2}<\frac{0.3x}{0.3}+\frac{0.2}{0.3}$$

Cancel out 0.3 and 0.3.

$$0.5x+\frac{3}{2}<x+\frac{0.2}{0.3}$$

Expand \frac{0.2}{0.3} by multiplying both numerator and the denominator by 10.

$$0.5x+\frac{3}{2}<x+\frac{2}{3}$$

Subtract x from both sides.

$$0.5x+\frac{3}{2}-x<\frac{2}{3}$$

Combine 0.5x and -x to get -0.5x.

$$-0.5x+\frac{3}{2}<\frac{2}{3}$$

Subtract \frac{3}{2} from both sides.

$$-0.5x<\frac{2}{3}-\frac{3}{2}$$

Least common multiple of 3 and 2 is 6. Convert \frac{2}{3} and \frac{3}{2} to fractions with denominator 6.

$$-0.5x<\frac{4}{6}-\frac{9}{6}$$

Since \frac{4}{6} and \frac{9}{6} have the same denominator, subtract them by subtracting their numerators.

$$-0.5x<\frac{4-9}{6}$$

Subtract 9 from 4 to get -5.

$$-0.5x<-\frac{5}{6}$$

Divide both sides by -0.5. Since -0.5 is negative, the inequality direction is changed.

$$x>\frac{-\frac{5}{6}}{-0.5}$$

Express \frac{-\frac{5}{6}}{-0.5} as a single fraction.

$$x>\frac{-5}{6\left(-0.5\right)}$$

Multiply 6 and -0.5 to get -3.

$$x>\frac{-5}{-3}$$

Fraction \frac{-5}{-3} can be simplified to \frac{5}{3} by removing the negative sign from both the numerator and the denominator.

$$x>\frac{5}{3}$$