4x - 8 \u2265 12 Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The solution set in interval notation is \u25a1 . (Simplify your answer. Use integers or fractions for any numbers in the expression. Type your answer in inter B. The solution set is \u2205. Graph the solution set on a number line. Choose the correct araph below
Answer (2)
Add 8 to both sides: 4 xg e q 20 .
Divide both sides by 4: xg e q 5 .
Express the solution in interval notation: [ 5 , ∞ ) .
The solution set is [ 5 , ∞ ) .
Explanation
Understanding the Inequality We are given the inequality 4 x − 8 g e q 12 . Our goal is to solve for x and express the solution in interval notation.
Isolating the x term First, we add 8 to both sides of the inequality to isolate the term with x :
4 x − 8 + 8 g e q 12 + 8
Simplifying the Inequality This simplifies to: 4 xg e q 20
Solving for x Next, we divide both sides of the inequality by 4 to solve for x :
4 4 x g e q 4 20
The Solution This simplifies to: xg e q 5
Interval Notation The solution set in interval notation is all real numbers greater than or equal to 5. This is written as [ 5 , ∞ ) .
Examples
Understanding inequalities is crucial in many real-world scenarios. For instance, imagine you're budgeting your expenses. You might want to ensure that your spending ( x ) plus your savings ($8) is always greater than or equal to your income ( 12 ) . T h e in e q u a l i t y x + 8 geq 12$ helps you determine the minimum amount you need to earn to meet your financial goals. Similarly, in manufacturing, companies use inequalities to ensure product quality, setting minimum and maximum acceptable values for dimensions or material properties.
The solution to the inequality 4 x − 8 ≥ 12 is x ≥ 5 , which can be expressed in interval notation as [ 5 , ∞ ) . To graph this, place a solid dot at 5 and shade to the right. Hence, the correct choice is [ 5 , ∞ ) .
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