Question
Solve the following inequality for \( r \). Write your answer in simplest form. \[ 7-(7 r+9)<-9 r+10-4 r \]
Ask by Ball Chambers. in the United States
Jan 25,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
\( r < 2 \)
Solution
To solve the inequality:
\[
7 - (7r + 9) < -9r + 10 - 4r
\]
**Step 1: Simplify Both Sides**
**Left Side:**
\[
7 - (7r + 9) = 7 - 7r - 9 = -7r - 2
\]
**Right Side:**
\[
-9r + 10 - 4r = -13r + 10
\]
Now, the inequality becomes:
\[
-7r - 2 < -13r + 10
\]
**Step 2: Rearrange the Inequality to Isolate \( r \)**
Add \( 13r \) to both sides:
\[
-7r + 13r - 2 < 10
\]
\[
6r - 2 < 10
\]
Add \( 2 \) to both sides:
\[
6r < 12
\]
Divide both sides by \( 6 \):
\[
r < 2
\]
**Final Answer:**
\[
r < 2
\]
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution The Deep Dive
To solve the inequality \( 7 - (7r + 9) < -9r + 10 - 4r \), we first simplify both sides. Starting with the left side: \[ 7 - (7r + 9) = 7 - 7r - 9 = -7r - 2 \] Now, simplifying the right side: \[ -9r + 10 - 4r = -9r - 4r + 10 = -13r + 10 \] So, we now have the inequality: \[ -7r - 2 < -13r + 10 \] Next, we can add \( 13r \) to both sides: \[ -7r + 13r - 2 < 10 \] This simplifies to: \[ 6r - 2 < 10 \] Now, we add \( 2 \) to both sides: \[ 6r < 10 + 2 \] This gives us: \[ 6r < 12 \] Finally, we divide both sides by \( 6 \): \[ r < 2 \] Thus, the solution in simplest form is: \[ \boxed{r < 2} \]
