To determine which graph shows all the values that satisfy the inequality [tex]\(\frac{2}{9} x + 3 > 4 \frac{5}{9}\)[/tex], we need to solve it step-by-step.
1. Simplify the inequality:
Start with the given inequality:
[tex]\[
\frac{2}{9} x + 3 > 4 \frac{5}{9}
\][/tex]
Convert the mixed number [tex]\(4 \frac{5}{9}\)[/tex] to an improper fraction.
[tex]\[
4 \frac{5}{9} = \frac{4 \cdot 9 + 5}{9} = \frac{36 + 5}{9} = \frac{41}{9}
\][/tex]
Substitute this back into the inequality:
[tex]\[
\frac{2}{9} x + 3 > \frac{41}{9}
\][/tex]
2. Isolate the variable [tex]\(x\)[/tex]:
Subtract 3 from both sides to begin isolating [tex]\(x\)[/tex]:
[tex]\[
\frac{2}{9} x + 3 - 3 > \frac{41}{9} - 3
\][/tex]
To subtract 3 from [tex]\(\frac{41}{9}\)[/tex], convert 3 to a fraction with the same denominator (9):
[tex]\[
3 = \frac{27}{9}
\][/tex]
So the inequality becomes:
[tex]\[
\frac{2}{9} x > \frac{41}{9} - \frac{27}{9}
\][/tex]
Simplify the right side:
[tex]\[
\frac{2}{9} x > \frac{41 - 27}{9} = \frac{14}{9}
\][/tex]
3. Solve for [tex]\(x\)[/tex]:
To isolate [tex]\(x\)[/tex], multiply both sides of the inequality by the reciprocal of [tex]\(\frac{2}{9}\)[/tex], which is [tex]\(\frac{9}{2}\)[/tex]:
[tex]\[
x > \frac{14}{9} \cdot \frac{9}{2}
\][/tex]
Simplify the right side:
[tex]\[
x > \frac{14 \cdot 9}{9 \cdot 2} = \frac{14}{2} = 7
\][/tex]
Therefore, the solution to the inequality is:
[tex]\[
x > 7
\][/tex]
The graph that represents all the values greater than 7 (i.e., all numbers to the right of 7 on a number line, not including 7 itself) would illustrate all [tex]\(x\)[/tex] such that [tex]\(x\)[/tex] is greater than [tex]\(7\)[/tex]. The correct graph will have an open circle at [tex]\(x = 7\)[/tex] (indicating that 7 itself is not included) and shading or an arrow indicating that all numbers greater than 7 satisfy the inequality.
