Solve 3/2(x-1)-2/3<x

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

Solve for x

$$x<\frac{13}{3}$$

Solution Steps

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

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

Multiply \frac{3}{2} and -1 to get -\frac{3}{2}.

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

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

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

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

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

Subtract 4 from -9 to get -13.

$$\frac{3}{2}x-\frac{13}{6}<x$$

Subtract x from both sides.

$$\frac{3}{2}x-\frac{13}{6}-x<0$$

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

$$\frac{1}{2}x-\frac{13}{6}<0$$

Add \frac{13}{6} to both sides. Anything plus zero gives itself.

$$\frac{1}{2}x<\frac{13}{6}$$

Multiply both sides by 2, the reciprocal of \frac{1}{2}. Since \frac{1}{2} is positive, the inequality direction remains the same.

$$x<\frac{13}{6}\times 2$$

Express \frac{13}{6}\times 2 as a single fraction.

$$x<\frac{13\times 2}{6}$$

Multiply 13 and 2 to get 26.

$$x<\frac{26}{6}$$

Reduce the fraction \frac{26}{6} to lowest terms by extracting and canceling out 2.

$$x<\frac{13}{3}$$