Solve the following inequality:
[tex]x \geq x-2(x+10)-7.5[/tex]
Answer (2)
Expand the right side of the inequality: x ≥ x − 2 x − 20 − 7.5 .
Simplify the inequality: x ≥ − x − 27.5 .
Add x to both sides: 2 x ≥ − 27.5 .
Divide by 2 to solve for x : x ≥ − 13.75 .
Explanation
Understanding the Problem We are given the inequality x ≥ x − 2 ( x + 10 ) − 7.5 Our goal is to solve for x , which means we want to isolate x on one side of the inequality to find the range of values that satisfy the given condition.
Expanding and Simplifying First, we need to expand the right side of the inequality by distributing the − 2 across the terms inside the parentheses: x ≥ x − 2 x − 20 − 7.5 Now, combine the like terms on the right side: x ≥ − x − 27.5
Isolating x Next, we want to isolate x . To do this, we add x to both sides of the inequality: x + x ≥ − x − 27.5 + x 2 x ≥ − 27.5
Solving for x Now, we divide both sides by 2 to solve for x :
2 2 x ≥ 2 − 27.5 x ≥ − 13.75
Examples
Imagine you're saving money and want to ensure you always have at least a certain amount. This inequality is like saying your current savings ( x ) must always be greater than or equal to your initial savings minus some expenses. Solving the inequality tells you the minimum amount you need to start with to meet your goal. For example, if you start with at least $ − 13.75 , even after accounting for expenses, you'll be on track.
To solve the inequality, we first expand and simplify it to find that x must be greater than or equal to − 13.75 . This means any value of x that is − 13.75 or larger will satisfy the inequality.
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