Solve the inequality. Express your answer using interval notation. Graph the solution set [tex]$\frac{x}{3} \geq 3-\frac{x}{9}$[/tex]
Answer (2)
Multiply both sides of the inequality by 9 to eliminate fractions: 3 x ≥ 27 − x .
Isolate x on one side of the inequality: 4 x ≥ 27 .
Solve for x: x ≥ 4 27 .
Express the solution in interval notation: [ 4 27 , ∞ ) .
Explanation
Understanding the Problem We are given the inequality 3 x ≥ 3 − 9 x . Our goal is to solve for x and express the solution in interval notation, then choose the correct graph representing the solution set.
Eliminating Fractions To solve the inequality, we first want to eliminate the fractions. We can do this by multiplying both sides of the inequality by the least common multiple of 3 and 9, which is 9. This gives us: 9 × 3 x ≥ 9 × ( 3 − 9 x ) 3 x ≥ 27 − x
Isolating x Next, we want to isolate x on one side of the inequality. We can add x to both sides: 3 x + x ≥ 27 − x + x 4 x ≥ 27
Solving for x Now, we can solve for x by dividing both sides by 4: 4 4 x ≥ 4 27 x ≥ 4 27
Interval Notation The solution to the inequality is x ≥ 4 27 . In interval notation, this is [ 4 27 , ∞ ) .
Graphing the Solution Set To graph the solution set, we need a number line with a closed bracket at 4 27 and an arrow extending to the right, indicating all values greater than or equal to 4 27 . Since 4 27 = 6.75 , we look for a graph with a closed bracket at 6.75 and an arrow extending to the right. Graph A matches this description.
Examples
Linear inequalities are useful in many real-world scenarios. For example, suppose you have a budget of $100 to spend on groceries. If you buy a certain number of items that cost $2 each, you can use an inequality to determine how many more items you can buy if another item costs $5. Solving such inequalities helps you manage your budget effectively. In this case, the inequality would be 2 x + 5 y ≤ 100 , where x is the number of 2 i t e m s an d y$ is the number of 5 i t e m s . S o l v in g f or y$ would tell you the maximum number of $5 items you can buy given a certain number of $2 items.
To solve the inequality 3 x ≥ 3 − 9 x , we find that the solution is x ≥ 4 27 . In interval notation, this is expressed as [ 4 27 , ∞ ) . The graph represents all values starting from 4 27 and extending towards infinity.
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