Start by distributing the 2 across the terms inside the parentheses on the left side of the inequality:
2(9−x)=2⋅9−2⋅x=18−2x 2(9-x) = 2 \cdot 9 - 2 \cdot x = 18 - 2x 2(9−x)=2⋅9−2⋅x=18−2x
So the inequality becomes:
18−2x>94x+1 18 - 2x > \frac{9}{4}x + 1 18−2x>49x+1
Add 2x2x2x to both sides to move all terms involving xxx to the right side:
18>94x+2x+1 18 > \frac{9}{4}x + 2x + 1 18>49x+2x+1
Combine the xxx terms on the right side:
18>(94+2)x+1 18 > \left(\frac{9}{4} + 2\right)x + 1 18>(49+2)x+1
Convert 2 to a fraction with a denominator of 4:
2=84 2 = \frac{8}{4} 2=48
So:
94+84=174 \frac{9}{4} + \frac{8}{4} = \frac{17}{4} 49+48=417
Thus, the inequality becomes:
18>174x+1 18 > \frac{17}{4}x + 1 18>417x+1
Subtract 1 from both sides to isolate the term with xxx:
18−1>174x 18 - 1 > \frac{17}{4}x 18−1>417x
17>174x 17 > \frac{17}{4}x 17>417x
Multiply both sides by 417\frac{4}{17}174 to solve for xxx:
x<417⋅17 x < \frac{4}{17} \cdot 17 x<174⋅17
x<4 x < 4 x<4
The solution to the inequality is:
x<4 \boxed{x < 4} x<4
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