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Chapter 3: Problem 42

Solve each inequality. $$5(r+3)-4(r-2)>7(r-4)-7 r$$

Short Answer

Expert verified
r > -51

Step by step solution

01

Expand the expressions

Expand the left and right sides of the inequality.Left side: \[5(r + 3) - 4(r - 2) = 5r + 15 - 4r + 8 = r + 23\]Right side: \[7(r - 4) - 7r = 7r - 28 - 7r = -28\]
02

Set up the simplified inequality

Using the simplified forms from step 1, set up the new inequality.\[r + 23 > -28\]
03

Isolate the variable 'r'

Subtract 23 from both sides to isolate 'r'.\[r + 23 - 23 > -28 - 23\]This simplifies to:\[r > -51\]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

algebra
Algebra is a branch of mathematics dealing with symbols and the rules for manipulating those symbols. In the context of solving inequalities, algebraic expressions involve variables and constants. These expressions need to be simplified to make the problem more manageable. For example, if you encounter an expression like 5(r + 3), you can apply the distributive property to expand it to 5r + 15, making it easier to handle in the next steps.
inequalities
Inequalities are mathematical statements that compare two values to show that one is less than, greater than, less than or equal to, or greater than or equal to the other. The inequality sign will dictate the relationship between the two sides. For instance, in \(5(r+3)-4(r-2)>7(r-4)-7r\), the '>' sign tells us that the left side is greater than the right side. To solve inequalities, we generally perform similar steps as we do with equations but with attention to the direction of the inequality sign. If we multiply or divide both sides by a negative number, the inequality sign must flip.
variable isolation
Variable isolation is a crucial step in solving equations and inequalities. The goal here is to get the variable by itself on one side of the inequality or equation. In our example, after simplifying both sides, we end up with an expression like \(r + 23 > -28\). To isolate r, subtract 23 from both sides. The result, \(r > -51\), tells us the range of values r can take. Always perform the same operation on both sides to maintain the inequality.
step-by-step solution
Solving inequalities step-by-step involves breaking down the problem into manageable chunks. Here鈥檚 how to handle our initial inequality:
1. **Expand the expressions: ** Distribute constants across terms inside parentheses:
\[5(r + 3) - 4(r - 2) = 5r + 15 - 4r + 8 = r + 23\] and \[7(r - 4) - 7r = 7r - 28 - 7r = -28\]
2. **Set up simplified inequality: ** Using the results from above steps, our inequality becomes: \r + 23 > -28 \.
3. **Isolate the variable: ** Subtract 23 from both sides to get \r > -51 \.
By following these steps, anyone can solve inequalities while understanding each stage of the process.

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Most popular questions from this chapter

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