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

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

Solve for x

$$x\leq \frac{79}{28}$$

Solution Steps

Use the distributive property to multiply 2 by 5-3x.

$$10-6x+\frac{1}{2}\geq \frac{-4}{3}\left(x+2\right)$$

Convert 10 to fraction \frac{20}{2}.

$$\frac{20}{2}-6x+\frac{1}{2}\geq \frac{-4}{3}\left(x+2\right)$$

Since \frac{20}{2} and \frac{1}{2} have the same denominator, add them by adding their numerators.

$$\frac{20+1}{2}-6x\geq \frac{-4}{3}\left(x+2\right)$$

Add 20 and 1 to get 21.

$$\frac{21}{2}-6x\geq \frac{-4}{3}\left(x+2\right)$$

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

$$\frac{21}{2}-6x\geq -\frac{4}{3}\left(x+2\right)$$

Use the distributive property to multiply -\frac{4}{3} by x+2.

$$\frac{21}{2}-6x\geq -\frac{4}{3}x-\frac{4}{3}\times 2$$

Express -\frac{4}{3}\times 2 as a single fraction.

$$\frac{21}{2}-6x\geq -\frac{4}{3}x+\frac{-4\times 2}{3}$$

Multiply -4 and 2 to get -8.

$$\frac{21}{2}-6x\geq -\frac{4}{3}x+\frac{-8}{3}$$

Fraction \frac{-8}{3} can be rewritten as -\frac{8}{3} by extracting the negative sign.

$$\frac{21}{2}-6x\geq -\frac{4}{3}x-\frac{8}{3}$$

Add \frac{4}{3}x to both sides.

$$\frac{21}{2}-6x+\frac{4}{3}x\geq -\frac{8}{3}$$

Combine -6x and \frac{4}{3}x to get -\frac{14}{3}x.

$$\frac{21}{2}-\frac{14}{3}x\geq -\frac{8}{3}$$

Subtract \frac{21}{2} from both sides.

$$-\frac{14}{3}x\geq -\frac{8}{3}-\frac{21}{2}$$

Least common multiple of 3 and 2 is 6. Convert -\frac{8}{3} and \frac{21}{2} to fractions with denominator 6.

$$-\frac{14}{3}x\geq -\frac{16}{6}-\frac{63}{6}$$

Since -\frac{16}{6} and \frac{63}{6} have the same denominator, subtract them by subtracting their numerators.

$$-\frac{14}{3}x\geq \frac{-16-63}{6}$$

Subtract 63 from -16 to get -79.

$$-\frac{14}{3}x\geq -\frac{79}{6}$$

Multiply both sides by -\frac{3}{14}, the reciprocal of -\frac{14}{3}. Since -\frac{14}{3} is negative, the inequality direction is changed.

$$x\leq -\frac{79}{6}\left(-\frac{3}{14}\right)$$

Multiply -\frac{79}{6} times -\frac{3}{14} by multiplying numerator times numerator and denominator times denominator.

$$x\leq \frac{-79\left(-3\right)}{6\times 14}$$

Do the multiplications in the fraction \frac{-79\left(-3\right)}{6\times 14}.

$$x\leq \frac{237}{84}$$

Reduce the fraction \frac{237}{84} to lowest terms by extracting and canceling out 3.

$$x\leq \frac{79}{28}$$