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: 3 x ≥ 27 − x .
Add x to both sides: 4 x ≥ 27 .
Divide both sides by 4: 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 , express the solution in interval notation, and 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: 9 ⋅ 3 x ≥ 9 ⋅ ( 3 − 9 x ) Simplifying, we get: 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 divide both sides by 4 to solve for x :
4 4 x ≥ 4 27 x ≥ 4 27
Expressing the Solution The solution to the inequality is x ≥ 4 27 . In interval notation, this is written as [ 4 27 , ∞ ) . Since 4 27 = 6.75 , we are looking for a graph that represents all values greater than or equal to 6.75.
Choosing the Correct Graph Comparing our solution to the given options, we need to choose the graph that shows a closed interval starting at 4 27 (or 6.75) and extending to infinity. Graph A is the correct graph.
Examples
Linear inequalities are used in various real-life scenarios. For example, suppose you have a budget of $100 to spend on groceries. If you buy items costing $20 and $30, you can use a linear inequality to determine how much more you can spend while staying within your budget. This helps in making informed decisions about resource allocation and constraints.
The solution to the inequality 3 x ≥ 3 − 9 x is x ≥ 4 27 , expressed in interval notation as [ 4 27 , ∞ ) . This indicates that x can be any value greater than or equal to 6.75. The graph of this solution includes a closed dot at 4 27 and a line extending to the right toward infinity.
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