Solve the Inequality for x (7-3x)/6+(5-2x)/3+(3-x)/2<1

$\frac{7 - 3 x}{6} + \frac{5 - 2 x}{3} + \frac{3 - x}{2} < 1$

Step 1

Simplify $\frac{7 - 3 x}{6} + \frac{5 - 2 x}{3} + \frac{3 - x}{2}$ .

Find the common denominator.

Multiply $\frac{5 - 2 x}{3}$ by $\frac{2}{2}$ .

$\frac{7 - 3 x}{6} + \frac{5 - 2 x}{3} \cdot \frac{2}{2} + \frac{3 - x}{2} < 1$

Multiply $\frac{5 - 2 x}{3}$ by $\frac{2}{2}$ .

$\frac{7 - 3 x}{6} + \frac{(5 - 2 x) \cdot 2}{3 \cdot 2} + \frac{3 - x}{2} < 1$

Multiply $\frac{3 - x}{2}$ by $\frac{3}{3}$ .

$\frac{7 - 3 x}{6} + \frac{(5 - 2 x) \cdot 2}{3 \cdot 2} + \frac{3 - x}{2} \cdot \frac{3}{3} < 1$

Multiply $\frac{3 - x}{2}$ by $\frac{3}{3}$ .

$\frac{7 - 3 x}{6} + \frac{(5 - 2 x) \cdot 2}{3 \cdot 2} + \frac{(3 - x) \cdot 3}{2 \cdot 3} < 1$

Reorder the factors of $3 \cdot 2$ .

$\frac{7 - 3 x}{6} + \frac{(5 - 2 x) \cdot 2}{2 \cdot 3} + \frac{(3 - x) \cdot 3}{2 \cdot 3} < 1$

Multiply $2$ by $3$ .

$\frac{7 - 3 x}{6} + \frac{(5 - 2 x) \cdot 2}{6} + \frac{(3 - x) \cdot 3}{2 \cdot 3} < 1$

Multiply $2$ by $3$ .

$\frac{7 - 3 x}{6} + \frac{(5 - 2 x) \cdot 2}{6} + \frac{(3 - x) \cdot 3}{6} < 1$

Combine the numerators over the common denominator.

$\frac{7 - 3 x + (5 - 2 x) \cdot 2 + (3 - x) \cdot 3}{6} < 1$

Simplify each term.

Apply the distributive property.

$\frac{7 - 3 x + 5 \cdot 2 - 2 x \cdot 2 + (3 - x) \cdot 3}{6} < 1$

Multiply $5$ by $2$ .

$\frac{7 - 3 x + 10 - 2 x \cdot 2 + (3 - x) \cdot 3}{6} < 1$

Multiply $2$ by $- 2$ .

$\frac{7 - 3 x + 10 - 4 x + (3 - x) \cdot 3}{6} < 1$

Apply the distributive property.

$\frac{7 - 3 x + 10 - 4 x + 3 \cdot 3 - x \cdot 3}{6} < 1$

Multiply $3$ by $3$ .

$\frac{7 - 3 x + 10 - 4 x + 9 - x \cdot 3}{6} < 1$

Multiply $3$ by $- 1$ .

$\frac{7 - 3 x + 10 - 4 x + 9 - 3 x}{6} < 1$

Simplify terms.

Add $7$ and $10$ .

$\frac{- 3 x + 17 - 4 x + 9 - 3 x}{6} < 1$

Subtract $4 x$ from $- 3 x$ .

$\frac{- 7 x + 17 + 9 - 3 x}{6} < 1$

Subtract $3 x$ from $- 7 x$ .

$\frac{- 10 x + 17 + 9}{6} < 1$

Add $17$ and $9$ .

$\frac{- 10 x + 26}{6} < 1$

Cancel the common factor of $- 10 x + 26$ and $6$ .

Factor $2$ out of $- 10 x$ .

$\frac{2 (- 5 x) + 26}{6} < 1$

Factor $2$ out of $26$ .

$\frac{2 (- 5 x) + 2 (13)}{6} < 1$

Factor $2$ out of $2 (- 5 x) + 2 (13)$ .

$\frac{2 (- 5 x + 13)}{6} < 1$

Cancel the common factors.

Factor $2$ out of $6$ .

$\frac{2 (- 5 x + 13)}{2 (3)} < 1$

Cancel the common factor.

$\frac{2 (- 5 x + 13)}{2 \cdot 3} < 1$

Rewrite the expression.

$\frac{- 5 x + 13}{3} < 1$

Factor $- 1$ out of $- 5 x$ .

$\frac{- (5 x) + 13}{3} < 1$

Rewrite $13$ as $- 1 (- 13)$ .

$\frac{- (5 x) - 1 (- 13)}{3} < 1$

Factor $- 1$ out of $- (5 x) - 1 (- 13)$ .

$\frac{- (5 x - 13)}{3} < 1$

Simplify the expression.

Rewrite $- (5 x - 13)$ as $- 1 (5 x - 13)$ .

$\frac{- 1 (5 x - 13)}{3} < 1$

Move the negative in front of the fraction.

$- \frac{5 x - 13}{3} < 1$

Step 2

Divide each term in $- \frac{5 x - 13}{3} < 1$ by $- 1$ and simplify.

Divide each term in $- \frac{5 x - 13}{3} < 1$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- \frac{5 x - 13}{3}}{- 1} > \frac{1}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{\frac{5 x - 13}{3}}{1} > \frac{1}{- 1}$

Divide $\frac{5 x - 13}{3}$ by $1$ .

$\frac{5 x - 13}{3} > \frac{1}{- 1}$

Simplify the right side.

Divide $1$ by $- 1$ .

$\frac{5 x - 13}{3} > - 1$

Step 3

Multiply both sides by $3$ .

$\frac{5 x - 13}{3} \cdot 3 > - 1 \cdot 3$

Step 4

Simplify.

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{5 x - 13}{3} \cdot 3 > - 1 \cdot 3$

Rewrite the expression.

$5 x - 13 > - 1 \cdot 3$

Simplify the right side.

Multiply $- 1$ by $3$ .

$5 x - 13 > - 3$

Step 5

Solve for $x$ .

Move all terms not containing $x$ to the right side of the inequality.

Add $13$ to both sides of the inequality.

$5 x > - 3 + 13$

Add $- 3$ and $13$ .

$5 x > 10$

Divide each term in $5 x > 10$ by $5$ and simplify.

Divide each term in $5 x > 10$ by $5$ .

$\frac{5 x}{5} > \frac{10}{5}$

Simplify the left side.

Cancel the common factor of $5$ .

Cancel the common factor.

$\frac{5 x}{5} > \frac{10}{5}$

Divide $x$ by $1$ .

$x > \frac{10}{5}$

Simplify the right side.

Divide $10$ by $5$ .

$x > 2$

Step 6

The result can be shown in multiple forms.

Inequality Form:

$x > 2$

Interval Notation:

$(2, \infty)$

Solve the Inequality for x (7-3x)/6+(5-2x)/3+(3-x)/2<1

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