Question
upstudy study bank question image url

Solve the following inequality. \[ (x-7)(x-1)^{2}(x+4)<0 \]

Ask by Griffin Lawson. in the United States
Mar 15,2025

Upstudy AI Solution

Tutor-Verified Answer

Answer

The solution to the inequality \((x-7)(x-1)^{2}(x+4)<0\) is \(x\) is between -4 and 1, and between 1 and 7.

Solution

Sign in to Unlock Answers for Free!

A Learning Platform Trusted by Millions of Real Students and Teachers.

star-icon Unlock

Answered by UpStudy AI and reviewed by a Professional Tutor

Extra Insights

To solve the inequality \((x-7)(x-1)^{2}(x+4)<0\), we first find the roots of the corresponding equation \((x-7)(x-1)^{2}(x+4) = 0\). The roots are: - \(x = 7\) - \(x = 1\) (with multiplicity 2) - \(x = -4\) These roots divide the number line into the following intervals: 1. \((- \infty, -4)\) 2. \((-4, 1)\) 3. \((1, 7)\) 4. \((7, +\infty)\) Next, we test each interval to see where the product \((x-7)(x-1)^{2}(x+4)\) is negative. 1. **Interval \((- \infty, -4)\)**: Choose \(x = -5\): \[ (-5-7)(-5-1)^{2}(-5+4) = (-12)(36)(-1) > 0 \quad (\text{positive}) \] 2. **Interval \((-4, 1)\)**: Choose \(x = 0\): \[ (0-7)(0-1)^{2}(0+4) = (-7)(1)(4) < 0 \quad (\text{negative}) \] 3. **Interval \((1, 7)\)**: Choose \(x = 2\): \[ (2-7)(2-1)^{2}(2+4) = (-5)(1)(6) < 0 \quad (\text{negative}) \] 4. **Interval \((7, +\infty)\)**: Choose \(x = 8\): \[ (8-7)(8-1)^{2}(8+4) = (1)(49)(12) > 0 \quad (\text{positive}) \] Now, we summarize what we found: - \((- \infty, -4)\): positive - \((-4, 1)\): negative - \((1, 7)\): negative - \((7, +\infty)\): positive Next, we check the critical points: - At \(x = -4\): \(((-4-7)(-4-1)^{2}(-4+4) = 0)\) - At \(x = 1\): \((1-7)(1-1)^{2}(1+4) = 0)\) - At \(x = 7\): \((7-7)(7-1)^{2}(7+4) = 0)\) Since we are looking for where \((x-7)(x-1)^{2}(x+4)<0\), we can conclude that the solution to the inequality is: \[ \boxed{(-4, 1) \cup (1, 7)} \]

Try Premium now!
Try Premium and ask Thoth AI unlimited math questions now!
Maybe later Go Premium
Study can be a real struggle
Why not UpStudy it?
Select your plan below
Premium

You can enjoy

Start now
  • Step-by-step explanations
  • 24/7 expert live tutors
  • Unlimited number of questions
  • No interruptions
  • Full access to Answer and Solution
  • Full Access to PDF Chat, UpStudy Chat, Browsing Chat
Basic

Totally free but limited

  • Limited Solution
Welcome to UpStudy!
Please sign in to continue the Thoth AI Chat journey
Continue with Email
Or continue with
By clicking “Sign in”, you agree to our Terms of Use & Privacy Policy