Which represents the solution of $-9.5+6 x \geq 42.1$ using interval notation?
A. ( $8.6, \infty$ )
B. [8.6, $\infty$ )
C. ( $\infty, 8.6$ )
D. ( $\infty, 8.6$ ]
Answer (1)
Add 9.5 to both sides: 6 x ≥ 51.6 .
Divide both sides by 6: x ≥ 8.6 .
Express the solution in interval notation: [ 8.6 , ∞ ) .
The solution to the inequality is [ 8.6 , ∞ ) .
Explanation
Understanding the Inequality We are given the inequality − 9.5 + 6 x ≥ 42.1 . Our goal is to solve for x and express the solution in interval notation.
Isolating the x term First, we add 9.5 to both sides of the inequality to isolate the term with x :
− 9.5 + 6 x + 9.5 ≥ 42.1 + 9.5 6 x ≥ 51.6
Solving for x Next, we divide both sides of the inequality by 6 to solve for x :
6 6 x ≥ 6 51.6 x ≥ 8.6
Expressing the Solution in Interval Notation The solution to the inequality is x ≥ 8.6 . In interval notation, this is represented as [ 8.6 , ∞ ) . The square bracket indicates that 8.6 is included in the solution, and the parenthesis indicates that infinity is not included.
Examples
Imagine you are saving money for a new bicycle that costs $42.1. You already have $9.5 saved, and you plan to save an additional 6 e a c h w ee k . T h e in e q u a l i t y -9.5 + 6x \geq 42.1 h e lp syo u d e t er min e h o w man y w ee k s ( x$) you need to save to afford the bicycle. Solving this inequality tells you that you need to save for at least 8.6 weeks, meaning you'll need to save for 9 full weeks to have enough money.
