Which graph shows all the values that satisfy [tex]$\frac{2}{9} x+3>4 \frac{5}{9}$[/tex]?
Answer (2)
Convert the mixed number to an improper fraction: 4 9 5 = 9 41 .
Subtract 3 from both sides: \frac{14}{9}"> 9 2 x > 9 14 .
Multiply both sides by 2 9 : 7"> x > 7 .
The solution is 7"> x > 7 , represented graphically by an open interval extending to positive infinity: 7}"> x > 7 .
Explanation
Understanding the Inequality We are given the inequality 4 \frac{5}{9}"> 9 2 x + 3 > 4 9 5 . Our goal is to isolate x to find the solution set.
Converting to Improper Fraction First, we convert the mixed number 4 9 5 to an improper fraction: 4 9 5 = 4 + 9 5 = 9 4 × 9 + 9 5 = 9 36 + 9 5 = 9 41 . So the inequality becomes \frac{41}{9}"> 9 2 x + 3 > 9 41 .
Subtracting 3 from Both Sides Next, we subtract 3 from both sides of the inequality. Since 3 = 9 27 , we have \frac{41}{9} - 3"> 9 2 x + 3 − 3 > 9 41 − 3 , which simplifies to \frac{41}{9} - \frac{27}{9} = \frac{41-27}{9} = \frac{14}{9}"> 9 2 x > 9 41 − 9 27 = 9 41 − 27 = 9 14 .
Multiplying by 9/2 Now, we multiply both sides of the inequality by 2 9 to isolate x : \frac{9}{2} \times \frac{14}{9}"> 2 9 × 9 2 x > 2 9 × 9 14 . This simplifies to \frac{9 \times 14}{2 \times 9} = \frac{14}{2} = 7"> x > 2 × 9 9 × 14 = 2 14 = 7 .
Finding the Solution Set Therefore, the solution to the inequality is 7"> x > 7 . This means that x can be any value greater than 7. On a number line, this is represented by an open circle at 7 and an arrow extending to the right, indicating all values greater than 7.
Examples
Understanding inequalities is crucial in many real-world scenarios. For instance, if you're budgeting your expenses, you might use an inequality to determine how much you can spend on entertainment each month while still covering your essential bills. Similarly, in science, inequalities are used to define acceptable ranges for experimental conditions, ensuring that results are valid and reliable. Inequalities also play a key role in optimization problems, helping to find the best possible outcome within given constraints.
To solve the inequality 4 \frac{5}{9}"> 9 2 x + 3 > 4 9 5 , we convert the mixed number to an improper fraction, isolate x , and find the solution is 7"> x > 7 . This means all values greater than 7 are solutions. Graphically, this is depicted on a number line with an open circle at 7 and an arrow extending right.
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