Solve (1)/(4)(x-2)+(2)/(3) < (1)/(6)(x-4) | Equatix - Math Solver

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Example

3(r+2s)=2t-4

a⋅(b-2)=3b

3(r+2s)=2t-4

a⋅(b-2)=3b

Question

$$\frac{ 1 }{ 4 } (x-2)+ \frac{ 2 }{ 3 } < \frac{ 1 }{ 6 } (x-4)$$

Solve for x

$x<-10$

Solution Steps

Use the distributive property to multiply $\frac{1}{4}$ by $x-2$.

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

Multiply $\frac{1}{4}$ and $-2$ to get $\frac{-2}{4}$.

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

Reduce the fraction $\frac{-2}{4}$ to lowest terms by extracting and canceling out $2$.

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

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

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

Since $-\frac{3}{6}$ and $\frac{4}{6}$ have the same denominator, add them by adding their numerators.

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

Add $-3$ and $4$ to get $1$.

$$\frac{1}{4}x+\frac{1}{6}<\frac{1}{6}\left(x-4\right)$$

Use the distributive property to multiply $\frac{1}{6}$ by $x-4$.

$$\frac{1}{4}x+\frac{1}{6}<\frac{1}{6}x+\frac{1}{6}\left(-4\right)$$

Multiply $\frac{1}{6}$ and $-4$ to get $\frac{-4}{6}$.

$$\frac{1}{4}x+\frac{1}{6}<\frac{1}{6}x+\frac{-4}{6}$$

Reduce the fraction $\frac{-4}{6}$ to lowest terms by extracting and canceling out $2$.

$$\frac{1}{4}x+\frac{1}{6}<\frac{1}{6}x-\frac{2}{3}$$

Subtract $\frac{1}{6}x$ from both sides.

$$\frac{1}{4}x+\frac{1}{6}-\frac{1}{6}x<-\frac{2}{3}$$

Combine $\frac{1}{4}x$ and $-\frac{1}{6}x$ to get $\frac{1}{12}x$.

$$\frac{1}{12}x+\frac{1}{6}<-\frac{2}{3}$$

Subtract $\frac{1}{6}$ from both sides.

$$\frac{1}{12}x<-\frac{2}{3}-\frac{1}{6}$$

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

$$\frac{1}{12}x<-\frac{4}{6}-\frac{1}{6}$$

Since $-\frac{4}{6}$ and $\frac{1}{6}$ have the same denominator, subtract them by subtracting their numerators.

$$\frac{1}{12}x<\frac{-4-1}{6}$$

Subtract $1$ from $-4$ to get $-5$.

$$\frac{1}{12}x<-\frac{5}{6}$$

Multiply both sides by $12$, the reciprocal of $\frac{1}{12}$. Since $\frac{1}{12}$ is positive, the inequality direction remains the same.

$$x<-\frac{5}{6}\times 12$$

Express $-\frac{5}{6}\times 12$ as a single fraction.

$$x<\frac{-5\times 12}{6}$$

Multiply $-5$ and $12$ to get $-60$.

$$x<\frac{-60}{6}$$

Divide $-60$ by $6$ to get $-10$.

$$x<-10$$