Solve the Inequality for z 1/2z+1/3z>=7/12z-1/4

$\frac{1}{2} z + \frac{1}{3} z \geq \frac{7}{12} z - \frac{1}{4}$

Simplify $\frac{1}{2} z + \frac{1}{3} z$ .

Simplify each term.

Combine $\frac{1}{2}$ and $z$ .

$\frac{z}{2} + \frac{1}{3} z \geq \frac{7}{12} z - \frac{1}{4}$

Combine $\frac{1}{3}$ and $z$ .

$\frac{z}{2} + \frac{z}{3} \geq \frac{7}{12} z - \frac{1}{4}$

To write $\frac{z}{2}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{z}{2} \cdot \frac{3}{3} + \frac{z}{3} \geq \frac{7}{12} z - \frac{1}{4}$

To write $\frac{z}{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{z}{2} \cdot \frac{3}{3} + \frac{z}{3} \cdot \frac{2}{2} \geq \frac{7}{12} z - \frac{1}{4}$

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{z}{2}$ by $\frac{3}{3}$ .

$\frac{z \cdot 3}{2 \cdot 3} + \frac{z}{3} \cdot \frac{2}{2} \geq \frac{7}{12} z - \frac{1}{4}$

Multiply $2$ by $3$ .

$\frac{z \cdot 3}{6} + \frac{z}{3} \cdot \frac{2}{2} \geq \frac{7}{12} z - \frac{1}{4}$

Multiply $\frac{z}{3}$ by $\frac{2}{2}$ .

$\frac{z \cdot 3}{6} + \frac{z \cdot 2}{3 \cdot 2} \geq \frac{7}{12} z - \frac{1}{4}$

Multiply $3$ by $2$ .

$\frac{z \cdot 3}{6} + \frac{z \cdot 2}{6} \geq \frac{7}{12} z - \frac{1}{4}$

Combine the numerators over the common denominator.

$\frac{z \cdot 3 + z \cdot 2}{6} \geq \frac{7}{12} z - \frac{1}{4}$

Simplify the numerator.

Move $3$ to the left of $z$ .

$\frac{3 \cdot z + z \cdot 2}{6} \geq \frac{7}{12} z - \frac{1}{4}$

Move $2$ to the left of $z$ .

$\frac{3 z + 2 \cdot z}{6} \geq \frac{7}{12} z - \frac{1}{4}$

Add $3 z$ and $2 z$ .

$\frac{5 z}{6} \geq \frac{7}{12} z - \frac{1}{4}$

Combine $\frac{7}{12}$ and $z$ .

$\frac{5 z}{6} \geq \frac{7 z}{12} - \frac{1}{4}$

Move all terms containing $z$ to the left side of the inequality.

Subtract $\frac{7 z}{12}$ from both sides of the inequality.

$\frac{5 z}{6} - \frac{7 z}{12} \geq - \frac{1}{4}$

To write $\frac{5 z}{6}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{5 z}{6} \cdot \frac{2}{2} - \frac{7 z}{12} \geq - \frac{1}{4}$

Write each expression with a common denominator of $12$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{5 z}{6}$ by $\frac{2}{2}$ .

$\frac{5 z \cdot 2}{6 \cdot 2} - \frac{7 z}{12} \geq - \frac{1}{4}$

Multiply $6$ by $2$ .

$\frac{5 z \cdot 2}{12} - \frac{7 z}{12} \geq - \frac{1}{4}$

Combine the numerators over the common denominator.

$\frac{5 z \cdot 2 - 7 z}{12} \geq - \frac{1}{4}$

Simplify the numerator.

Factor $z$ out of $5 z \cdot 2 - 7 z$ .

Factor $z$ out of $5 z \cdot 2$ .

$\frac{z (5 \cdot 2) - 7 z}{12} \geq - \frac{1}{4}$

Factor $z$ out of $- 7 z$ .

$\frac{z (5 \cdot 2) + z \cdot - 7}{12} \geq - \frac{1}{4}$

Factor $z$ out of $z (5 \cdot 2) + z \cdot - 7$ .

$\frac{z (5 \cdot 2 - 7)}{12} \geq - \frac{1}{4}$

Multiply $5$ by $2$ .

$\frac{z (10 - 7)}{12} \geq - \frac{1}{4}$

Subtract $7$ from $10$ .

$\frac{z \cdot 3}{12} \geq - \frac{1}{4}$

Cancel the common factor of $3$ and $12$ .

Factor $3$ out of $z \cdot 3$ .

$\frac{3 \cdot z}{12} \geq - \frac{1}{4}$

Cancel the common factors.

Factor $3$ out of $12$ .

$\frac{3 \cdot z}{3 \cdot 4} \geq - \frac{1}{4}$

Cancel the common factor.

$\frac{3 \cdot z}{3 \cdot 4} \geq - \frac{1}{4}$

Rewrite the expression.

$\frac{z}{4} \geq - \frac{1}{4}$

Since the expression on each side of the equation has the same denominator, the numerators must be equal.

$z \geq - 1$

The result can be shown in multiple forms.

Inequality Form:

$z \geq - 1$

Interval Notation:

$[- 1, \infty)$

Solve the Inequality for z 1/2z+1/3z>=7/12z-1/4

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