Solve the inequality.
[tex]2(4 x-3) \geq-3(3 x)+5 x[/tex]
A. [tex]x \geq 0.5[/tex]
B. [tex]x \geq 2[/tex]
C. [tex](-\infty, 0.5][/tex]
D. [tex](-\infty, 2][/tex]
Answer (2)
Expand both sides of the inequality: 2 ( 4 x − 3 ) ≥ − 3 ( 3 x ) + 5 x becomes 8 x − 6 ≥ − 9 x + 5 x .
Simplify the inequality: 8 x − 6 ≥ − 4 x .
Isolate x by adding 4 x and 6 to both sides: 12 x ≥ 6 .
Solve for x by dividing both sides by 12 : x ≥ 0.5 . The solution is x ≥ 0.5 .
Explanation
Understanding the Inequality We are given the inequality 2 ( 4 x − 3 ) ≥ − 3 ( 3 x ) + 5 x . Our goal is to solve for x , which means we want to isolate x on one side of the inequality.
Expanding Both Sides First, we expand both sides of the inequality: 2 ( 4 x − 3 ) ≥ − 3 ( 3 x ) + 5 x 8 x − 6 ≥ − 9 x + 5 x 8 x − 6 ≥ − 4 x
Isolating x Terms Next, we want to isolate the x terms. We can add 4 x to both sides of the inequality: 8 x − 6 + 4 x ≥ − 4 x + 4 x 12 x − 6 ≥ 0
Further Isolating x Now, we add 6 to both sides: 12 x − 6 + 6 ≥ 0 + 6 12 x ≥ 6
Solving for x Finally, we divide both sides by 12: 12 12 x ≥ 12 6 x ≥ 2 1 x ≥ 0.5
Expressing the Solution The solution to the inequality is x ≥ 0.5 . This means that x can be any value greater than or equal to 0.5. In interval notation, this is written as [ 0.5 , ∞ ) .
Examples
Imagine you're saving money for a new video game that costs $60. You start with $10 and plan to save $5 each week. This inequality helps you determine how many weeks you need to save to have at least $60. By solving a similar inequality, you can find the minimum number of weeks required to reach your savings goal. Understanding inequalities is crucial for budgeting, financial planning, and making informed decisions about your resources.
The solution to the inequality 2 ( 4 x − 3 ) ≥ − 3 ( 3 x ) + 5 x is x ≥ 0.5 . Therefore, the answer is option A: x ≥ 0.5 .
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