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

Transcript text: Complete the solution for the inequality $2(9-x)>\frac{9}{4} x+1$.

Solution

Step 1: Distribute the 2 on the left side

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:

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

Step 2: Move all terms involving $x$ to one side

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:

\[ 18 > \frac{17}{4}x + 1 \]

Step 3: Isolate the term with $x$

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

\[ 18 - 1 > \frac{17}{4}x \]

\[ 17 > \frac{17}{4}x \]

Step 4: Solve for $x$

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

\[ x < \frac{4}{17} \cdot 17 \]

\[ x < 4 \]

The solution to the inequality is:

\[ \boxed{x < 4} \]

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