Solve the inequality.
[tex]2(4+2 x) \geq 5 x+5[/tex]
A. [tex]x \leq-2[/tex]
B. [tex]x \geq-2[/tex]
C. [tex]x \leq 3[/tex]
D. [tex]x \geq 3[/tex]
Answer (2)
The solution to the inequality 2 ( 4 + 2 x ) ≥ 5 x + 5 is x ≤ 3 . This means that values of x that are less than or equal to 3 satisfy the inequality. The correct answer choice is C.
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Expand the left side of the inequality: 2 ( 4 + 2 x ) g e 5 x + 5 becomes 8 + 4 xg e 5 x + 5 .
Subtract 4 x from both sides: 8 g e x + 5 .
Subtract 5 from both sides: 3 g e x .
Rewrite the inequality: x ≤ 3 . The solution is x ≤ 3 .
Explanation
Understanding the Inequality We are given the inequality 2 ( 4 + 2 x ) g e q 5 x + 5 . Our goal is to find the values of x that satisfy this inequality. We will simplify the inequality step by step to isolate x on one side.
Expanding the Left Side First, we expand the left side of the inequality by distributing the 2: $2(4+2x) = 2
imes 4 + 2
imes 2x = 8 + 4x$. So the inequality becomes 8 + 4 xg e q 5 x + 5 .
Isolating x Terms Next, we want to get all the x terms on one side of the inequality. Let's subtract 4 x from both sides: 8 + 4 x − 4 xg e q 5 x + 5 − 4 x , which simplifies to 8 g e q x + 5 .
Isolating x Now, we want to isolate x completely. We subtract 5 from both sides of the inequality: 8 − 5 g e q x + 5 − 5 , which simplifies to 3 g e q x .
Rewriting the Inequality Finally, we can rewrite the inequality 3 g e q x as x ≤ 3 . This means that x must be less than or equal to 3.
Final Answer Comparing our solution x ≤ 3 with the given options, we see that the correct answer is x ≤ 3 .
Examples
Imagine you're baking cookies and the recipe says you need at least twice the amount of flour as sugar, plus an extra amount. Solving inequalities like this helps you figure out the possible amounts of sugar you can use while still following the recipe's rule. Understanding inequalities is useful in many real-life situations, such as managing budgets, planning projects, or even optimizing resources to meet certain constraints. This algebraic approach ensures that you stay within the required limits while making the most of available resources.
