Which values of [tex]$x$[/tex] satisfy the inequality [tex]$0.5 x-0.75 \geq 3.25$[/tex]?
A. [tex]$x \geq 5$[/tex]
B. [tex]$x \leq 5$[/tex]
C. [tex]$x \leq 8$[/tex]
D. [tex]$x \geq 8$[/tex]
Answer (2)
The inequality 0.5 x − 0.75 ≥ 3.25 simplifies to x ≥ 8 after performing the necessary algebraic steps. The choice that corresponds to this solution is D. x ≥ 8 .
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Add 0.75 to both sides of the inequality: 0.5 x − 0.75 + 0.75 ≥ 3.25 + 0.75 .
Simplify: 0.5 x ≥ 4 .
Multiply both sides by 2: 2 ( 0.5 x ) ≥ 2 ( 4 ) .
Solve for x : x ≥ 8 . The solution is x ≥ 8 .
Explanation
Understanding the Inequality We are given the inequality 0.5 x − 0.75 ≥ 3.25 . Our goal is to isolate x to find the values that satisfy this inequality.
Adding to Both Sides First, we add 0.75 to both sides of the inequality to eliminate the constant term on the left side: 0.5 x − 0.75 + 0.75 ≥ 3.25 + 0.75
Simplifying the Inequality This simplifies to: 0.5 x ≥ 4
Multiplying to Isolate x Next, we multiply both sides of the inequality by 2 to solve for x :
2 ( 0.5 x ) ≥ 2 ( 4 )
Final Solution This simplifies to: x ≥ 8 So, the values of x that satisfy the inequality are all values greater than or equal to 8.
Examples
Imagine you're saving money for a new video game that costs $3.25, but you already owe your friend $0.75. If you earn $0.50 per chore, this problem helps you determine how many chores you need to do to buy the game and pay back your friend. Understanding inequalities helps you manage your earnings and expenses effectively in real life. This algebraic approach ensures you meet your financial goals.
