Which graph shows the solution set of the inequality $2.9(x+8)<26.1$?
Answer (2)
The solution to the inequality 2.9 ( x + 8 ) < 26.1 is x < 1 , meaning all real numbers less than 1 satisfy the inequality. This is represented graphically with an open point at 1 and shading to the left. Thus, the graph of the solution set shows all values less than 1 .
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Divide both sides of the inequality by 2.9 : x + 8 < 9 .
Subtract 8 from both sides: x < 1 .
The solution set includes all real numbers less than 1 .
The solution to the inequality is x < 1 .
Explanation
Understanding the Inequality We are given the inequality 2.9 ( x + 8 ) < 26.1 . Our goal is to find the solution set for x , which means we want to isolate x on one side of the inequality.
Dividing Both Sides First, we divide both sides of the inequality by 2.9 :
2.9 2.9 ( x + 8 ) < 2.9 26.1 This simplifies to: x + 8 < 9
Subtracting 8 Next, we subtract 8 from both sides of the inequality: x + 8 − 8 < 9 − 8 This simplifies to: x < 1
The Solution Set The solution set is all real numbers less than 1 . On a number line, this is represented by an open interval extending to the left from 1 .
Final Answer Therefore, the solution to the inequality 2.9 ( x + 8 ) < 26.1 is x < 1 .
Examples
Understanding inequalities is crucial in many real-world scenarios. For instance, imagine you're budgeting for a project and need to ensure that your expenses don't exceed a certain limit. Inequalities help you model these constraints and determine the range of possible spending while staying within budget. Similarly, in science, inequalities are used to define acceptable ranges for experimental conditions, ensuring that results are valid and reliable. This problem demonstrates a fundamental skill in mathematical modeling and problem-solving.
