Question
Solve the following inequality. Write the inequality in interval notation, and graph it. \( 7 r+2 \geq 3 r-26 \) The solution set is (Type your answer in interval notation.)
Ask by Carrillo Mills. in the United States
Feb 27,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The solution is \( r \geq -7 \) in interval notation: \([-7, \infty)\).
Solution
To solve the inequality 7r + 2 ≥ 3r - 26, follow these steps:
1. Subtract 3r from both sides:
7r - 3r + 2 ≥ (3r - 3r) - 26
4r + 2 ≥ -26
2. Subtract 2 from both sides:
4r ≥ -26 - 2
4r ≥ -28
3. Divide both sides by 4:
r ≥ -7
The solution in interval notation is: [-7, ∞).
Graphically, on a number line you would mark a closed circle at -7 and shade all numbers to the right of -7:
<---(-10)---(-7)●============→
This shows that the inequality holds for all values of r that are greater than or equal to -7.
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Simplify this solution Beyond the Answer
First, let's solve the inequality step by step. We start with the original inequality: \( 7r + 2 \geq 3r - 26 \) Subtract \( 3r \) from both sides: \( 7r - 3r + 2 \geq -26 \) This simplifies to: \( 4r + 2 \geq -26 \) Next, we subtract 2 from both sides: \( 4r \geq -28 \) Now, divide both sides by 4: \( r \geq -7 \) In interval notation, this is expressed as: \[ [-7, \infty) \] To graph this inequality on a number line, you would draw a solid circle at \(-7\) (indicating that \(-7\) is included in the solution) and shade the line to the right, toward positive infinity to indicate all numbers greater than or equal to \(-7\) are included in the solution set.
