To solve the inequality [tex]\(\frac{2}{9} x + 3 > 4 \frac{5}{9}\)[/tex], we will follow these steps systematically:
1. Convert the mixed number:
The right-hand side of the inequality, [tex]\(4 \frac{5}{9}\)[/tex], needs to be converted into an improper fraction.
[tex]\[
4 \frac{5}{9} = 4 + \frac{5}{9}
\][/tex]
[tex]\[
= \frac{36}{9} + \frac{5}{9} = \frac{41}{9}
\][/tex]
So, the inequality now looks like:
[tex]\[
\frac{2}{9} x + 3 > \frac{41}{9}
\][/tex]
2. Isolate the term containing [tex]\( x \)[/tex]:
To isolate [tex]\( x \)[/tex], we first subtract 3 from both sides of the inequality. Note that 3 can be written as [tex]\(\frac{27}{9}\)[/tex] to have a common denominator.
[tex]\[
\frac{2}{9} x + 3 > \frac{41}{9}
\][/tex]
[tex]\[
\frac{2}{9} x + 3 - 3 > \frac{41}{9} - 3
\][/tex]
[tex]\[
\frac{2}{9} x > \frac{41}{9} - \frac{27}{9}
\][/tex]
[tex]\[
\frac{2}{9} x > \frac{14}{9}
\][/tex]
3. Solve for [tex]\( x \)[/tex]:
To solve for [tex]\( x \)[/tex], we need to eliminate the fraction by multiplying both sides by the reciprocal of [tex]\(\frac{2}{9}\)[/tex], which is [tex]\(\frac{9}{2}\)[/tex].
[tex]\[
x > \frac{14}{9} \times \frac{9}{2}
\][/tex]
[tex]\[
x > 7
\][/tex]
Thus, the solution to the inequality is [tex]\( x > 7 \)[/tex].
### Graphing the Solution:
To graph the inequality [tex]\( x > 7 \)[/tex], you would draw a number line:
1. Draw a number line and mark the point corresponding to [tex]\( x = 7 \)[/tex].
2. Since the inequality is strictly greater than (i.e., does not include the boundary point), place an open circle at [tex]\( x = 7 \)[/tex].
3. Shade the region to the right of 7 to indicate all the numbers greater than 7.
In summary, the graph that shows all the values that satisfy [tex]\(\frac{2}{9} x + 3 > 4 \frac{5}{9}\)[/tex] is a number line with an open circle at [tex]\( x = 7 \)[/tex] and shading extending to the right of [tex]\( x = 7 \)[/tex].
