Solve 1/2(x-5)leq1/3(2x-1)

$$\frac { 1 } { 2 } ( x - 5 ) \leq \frac { 1 } { 3 } ( 2 x - 1 )$$

Solve for x

$$x\geq -13$$

Solution Steps

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

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

Multiply \frac{1}{2} and -5 to get \frac{-5}{2}.

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

Fraction \frac{-5}{2} can be rewritten as -\frac{5}{2} by extracting the negative sign.

$$\frac{1}{2}x-\frac{5}{2}\leq \frac{1}{3}\left(2x-1\right)$$

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

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

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

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

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

$$\frac{1}{2}x-\frac{5}{2}\leq \frac{2}{3}x-\frac{1}{3}$$

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

$$\frac{1}{2}x-\frac{5}{2}-\frac{2}{3}x\leq -\frac{1}{3}$$

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

$$-\frac{1}{6}x-\frac{5}{2}\leq -\frac{1}{3}$$

Add \frac{5}{2} to both sides.

$$-\frac{1}{6}x\leq -\frac{1}{3}+\frac{5}{2}$$

Least common multiple of 3 and 2 is 6. Convert -\frac{1}{3} and \frac{5}{2} to fractions with denominator 6.

$$-\frac{1}{6}x\leq -\frac{2}{6}+\frac{15}{6}$$

Since -\frac{2}{6} and \frac{15}{6} have the same denominator, add them by adding their numerators.

$$-\frac{1}{6}x\leq \frac{-2+15}{6}$$

Add -2 and 15 to get 13.

$$-\frac{1}{6}x\leq \frac{13}{6}$$

Multiply both sides by -6, the reciprocal of -\frac{1}{6}. Since -\frac{1}{6} is negative, the inequality direction is changed.

$$x\geq \frac{13}{6}\left(-6\right)$$

Express \frac{13}{6}\left(-6\right) as a single fraction.

$$x\geq \frac{13\left(-6\right)}{6}$$

Multiply 13 and -6 to get -78.

$$x\geq \frac{-78}{6}$$

Divide -78 by 6 to get -13.

$$x\geq -13$$