Solve (3/2(x-2)]-(x+4)>(1/3(2x+5)]+x

$$( \frac { 3 } { 2 } ( x - 2 ) ] - ( x + 4 ) > ( \frac { 1 } { 3 } ( 2 x + 5 ) ] + x$$

Solve for x

$$x<-\frac{52}{7}$$

Solution Steps

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

$$\frac{3}{2}x+\frac{3}{2}\left(-2\right)-\left(x+4\right)>\frac{1}{3}\left(2x+5\right)+x$$

Express \frac{3}{2}\left(-2\right) as a single fraction.

$$\frac{3}{2}x+\frac{3\left(-2\right)}{2}-\left(x+4\right)>\frac{1}{3}\left(2x+5\right)+x$$

Multiply 3 and -2 to get -6.

$$\frac{3}{2}x+\frac{-6}{2}-\left(x+4\right)>\frac{1}{3}\left(2x+5\right)+x$$

Divide -6 by 2 to get -3.

$$\frac{3}{2}x-3-\left(x+4\right)>\frac{1}{3}\left(2x+5\right)+x$$

To find the opposite of x+4, find the opposite of each term.

$$\frac{3}{2}x-3-x-4>\frac{1}{3}\left(2x+5\right)+x$$

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

$$\frac{1}{2}x-3-4>\frac{1}{3}\left(2x+5\right)+x$$

Subtract 4 from -3 to get -7.

$$\frac{1}{2}x-7>\frac{1}{3}\left(2x+5\right)+x$$

Use the distributive property to multiply \frac{1}{3} by 2x+5.

$$\frac{1}{2}x-7>\frac{1}{3}\times 2x+\frac{1}{3}\times 5+x$$

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

$$\frac{1}{2}x-7>\frac{2}{3}x+\frac{1}{3}\times 5+x$$

Multiply \frac{1}{3} and 5 to get \frac{5}{3}.

$$\frac{1}{2}x-7>\frac{2}{3}x+\frac{5}{3}+x$$

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

$$\frac{1}{2}x-7>\frac{5}{3}x+\frac{5}{3}$$

Subtract \frac{5}{3}x from both sides.

$$\frac{1}{2}x-7-\frac{5}{3}x>\frac{5}{3}$$

Combine \frac{1}{2}x and -\frac{5}{3}x to get -\frac{7}{6}x.

$$-\frac{7}{6}x-7>\frac{5}{3}$$

Add 7 to both sides.

$$-\frac{7}{6}x>\frac{5}{3}+7$$

Convert 7 to fraction \frac{21}{3}.

$$-\frac{7}{6}x>\frac{5}{3}+\frac{21}{3}$$

Since \frac{5}{3} and \frac{21}{3} have the same denominator, add them by adding their numerators.

$$-\frac{7}{6}x>\frac{5+21}{3}$$

Add 5 and 21 to get 26.

$$-\frac{7}{6}x>\frac{26}{3}$$

Multiply both sides by -\frac{6}{7}, the reciprocal of -\frac{7}{6}. Since -\frac{7}{6} is negative, the inequality direction is changed.

$$x<\frac{26}{3}\left(-\frac{6}{7}\right)$$

Multiply \frac{26}{3} times -\frac{6}{7} by multiplying numerator times numerator and denominator times denominator.

$$x<\frac{26\left(-6\right)}{3\times 7}$$

Do the multiplications in the fraction \frac{26\left(-6\right)}{3\times 7}.

$$x<\frac{-156}{21}$$

Reduce the fraction \frac{-156}{21} to lowest terms by extracting and canceling out 3.

$$x<-\frac{52}{7}$$