Solve the inequality:
[tex]9-4 r\ \textgreater \ 5[/tex]
A. [tex]r\ \textgreater \ 1[/tex]
B. [tex]r\ \textless \ -4[/tex]
C. [tex]r\ \textless \ 1[/tex]
D. [tex]r\ \textless \ -1[/tex]
Answer (2)
Subtract 9 from both sides: -4"> − 4 r > − 4 .
Divide both sides by -4 and flip the inequality sign: r < 1 .
The solution to the inequality is r < 1 .
The correct answer is r < 1 , which corresponds to option C. r < 1
Explanation
Understanding the Inequality We are given the inequality 5"> 9 − 4 r > 5 . Our goal is to isolate r on one side of the inequality to find the solution.
Subtracting 9 from Both Sides First, we subtract 9 from both sides of the inequality to start isolating the term with r :
5 - 9"> 9 − 4 r − 9 > 5 − 9
This simplifies to:
-4"> − 4 r > − 4
Dividing by -4 and Reversing the Inequality Next, we divide both sides of the inequality by -4. Remember that when we divide or multiply an inequality by a negative number, we must reverse the direction of the inequality sign:
− 4 − 4 r < − 4 − 4
This simplifies to:
r < 1
Finding the Correct Option The solution to the inequality is r < 1 . Now we compare this solution to the given options to find the correct answer.
Examples
Understanding inequalities is crucial in many real-world scenarios. For example, imagine you're managing a budget and need to ensure your expenses ( 4 r ) plus a fixed cost ($5) remain less than your total income ($9). Solving the inequality 5"> 9 − 4 r > 5 helps you determine the maximum amount you can spend on each item (represented by r ) to stay within your budget. This kind of problem-solving applies to resource allocation, financial planning, and even setting limits in games or sports.
To solve the inequality 5"> 9 − 4 r > 5 , we isolate r to find that r < 1 . Therefore, the correct answer is option C. This means that any value of r less than 1 satisfies the inequality.
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