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

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

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Solution Steps

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(9x)=292x=182x 2(9-x) = 2 \cdot 9 - 2 \cdot x = 18 - 2x

So the inequality becomes:

182x>94x+1 18 - 2x > \frac{9}{4}x + 1

Step 2: Move all terms involving xx to one side

Add 2x2x to both sides to move all terms involving xx to the right side:

18>94x+2x+1 18 > \frac{9}{4}x + 2x + 1

Combine the xx terms on the right side:

18>(94+2)x+1 18 > \left(\frac{9}{4} + 2\right)x + 1

Convert 2 to a fraction with a denominator of 4:

2=84 2 = \frac{8}{4}

So:

94+84=174 \frac{9}{4} + \frac{8}{4} = \frac{17}{4}

Thus, the inequality becomes:

18>174x+1 18 > \frac{17}{4}x + 1

Step 3: Isolate the term with xx

Subtract 1 from both sides to isolate the term with xx:

181>174x 18 - 1 > \frac{17}{4}x

17>174x 17 > \frac{17}{4}x

Step 4: Solve for xx

Multiply both sides by 417\frac{4}{17} to solve for xx:

x<41717 x < \frac{4}{17} \cdot 17

x<4 x < 4

Final Answer

The solution to the inequality is:

x<4 \boxed{x < 4}

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