Questions: Complete the solution for the inequality 2(9-x) > 9/4 x + 1.

Solution Steps

Start by distributing the 2 across the terms inside the parentheses on the left side of the inequality:

$$2(9-x) = 2 \cdot 9 - 2 \cdot x = 18 - 2x$$

So the inequality becomes:

$$18 - 2x > \frac{9}{4}x + 1$$

Add $2x$ to both sides to move all terms involving $x$ to the right side:

$$18 > \frac{9}{4}x + 2x + 1$$

Combine the $x$ terms on the right side:

$$18 > \left(\frac{9}{4} + 2\right)x + 1$$

Convert 2 to a fraction with a denominator of 4:

$$2 = \frac{8}{4}$$

So:

$$\frac{9}{4} + \frac{8}{4} = \frac{17}{4}$$

Thus, the inequality becomes:

$$18 > \frac{17}{4}x + 1$$

Subtract 1 from both sides to isolate the term with $x$:

$$18 - 1 > \frac{17}{4}x$$

$$17 > \frac{17}{4}x$$

Multiply both sides by $\frac{4}{17}$ to solve for $x$:

$$x < \frac{4}{17} \cdot 17$$

$$x < 4$$

Final Answer