Solve 3/4x-2/3<1/2(x-2)

$$\frac { 3 } { 4 } x - \frac { 2 } { 3 } < \frac { 1 } { 2 } ( x - 2 )$$

Solve for x

$$x<-\frac{4}{3}$$

Solution Steps

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

$$\frac{3}{4}x-\frac{2}{3}<\frac{1}{2}x+\frac{1}{2}\left(-2\right)$$

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

$$\frac{3}{4}x-\frac{2}{3}<\frac{1}{2}x+\frac{-2}{2}$$

Divide -2 by 2 to get -1.

$$\frac{3}{4}x-\frac{2}{3}<\frac{1}{2}x-1$$

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

$$\frac{3}{4}x-\frac{2}{3}-\frac{1}{2}x<-1$$

Combine \frac{3}{4}x and -\frac{1}{2}x to get \frac{1}{4}x.

$$\frac{1}{4}x-\frac{2}{3}<-1$$

Add \frac{2}{3} to both sides.

$$\frac{1}{4}x<-1+\frac{2}{3}$$

Convert -1 to fraction -\frac{3}{3}.

$$\frac{1}{4}x<-\frac{3}{3}+\frac{2}{3}$$

Since -\frac{3}{3} and \frac{2}{3} have the same denominator, add them by adding their numerators.

$$\frac{1}{4}x<\frac{-3+2}{3}$$

Add -3 and 2 to get -1.

$$\frac{1}{4}x<-\frac{1}{3}$$

Multiply both sides by 4, the reciprocal of \frac{1}{4}. Since \frac{1}{4} is positive, the inequality direction remains the same.

$$x<-\frac{1}{3}\times 4$$

Express -\frac{1}{3}\times 4 as a single fraction.

$$x<\frac{-4}{3}$$

Fraction \frac{-4}{3} can be rewritten as -\frac{4}{3} by extracting the negative sign.

$$x<-\frac{4}{3}$$