Solve for x x+3/4<-x/2+9/4

$x + \frac{3}{4} < - \frac{x}{2} + \frac{9}{4}$

Step 1

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

Add $\frac{x}{2}$ to both sides of the inequality.

$x + \frac{3}{4} + \frac{x}{2} < \frac{9}{4}$

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

$x \cdot \frac{2}{2} + \frac{x}{2} + \frac{3}{4} < \frac{9}{4}$

Combine $x$ and $\frac{2}{2}$ .

$\frac{x \cdot 2}{2} + \frac{x}{2} + \frac{3}{4} < \frac{9}{4}$

Combine the numerators over the common denominator.

$\frac{x \cdot 2 + x}{2} + \frac{3}{4} < \frac{9}{4}$

Add $x \cdot 2$ and $x$ .

Reorder $x$ and $2$ .

$\frac{2 \cdot x + x}{2} + \frac{3}{4} < \frac{9}{4}$

Add $2 \cdot x$ and $x$ .

$\frac{3 \cdot x}{2} + \frac{3}{4} < \frac{9}{4}$

$\frac{3 x}{2} + \frac{3}{4} < \frac{9}{4}$

Step 2

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

Subtract $\frac{3}{4}$ from both sides of the inequality.

$\frac{3 x}{2} < \frac{9}{4} - \frac{3}{4}$

Combine the numerators over the common denominator.

$\frac{3 x}{2} < \frac{9 - 3}{4}$

Subtract $3$ from $9$ .

$\frac{3 x}{2} < \frac{6}{4}$

Cancel the common factor of $6$ and $4$ .

Factor $2$ out of $6$ .

$\frac{3 x}{2} < \frac{2 (3)}{4}$

Cancel the common factors.

Factor $2$ out of $4$ .

$\frac{3 x}{2} < \frac{2 \cdot 3}{2 \cdot 2}$

Cancel the common factor.

$\frac{3 x}{2} < \frac{2 \cdot 3}{2 \cdot 2}$

Rewrite the expression.

$\frac{3 x}{2} < \frac{3}{2}$

Step 3

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

$3 x < 3$

Step 4

Divide each term in $3 x < 3$ by $3$ and simplify.

Divide each term in $3 x < 3$ by $3$ .

$\frac{3 x}{3} < \frac{3}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 x}{3} < \frac{3}{3}$

Divide $x$ by $1$ .

$x < \frac{3}{3}$

Simplify the right side.

Divide $3$ by $3$ .

$x < 1$

Step 5

The result can be shown in multiple forms.

Inequality Form:

$x < 1$

Interval Notation:

$(- \infty, 1)$

Solve for x x+3/4<-x/2+9/4

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