ESTION 1 Given: $$ T=\frac{\sqrt{x+5}}{x-1} $$, where $$ x \in\{-6 ;-2 ; 1 ; 4\} $$. Choose from the given set of numbers a value of $$ x $$ for which $$ T $$ will be: 1.I.1 rational 1.1.2 non-real

Question

UpStudy AI · Accepted Answer

**Step 1. Determine the domain of the function $$ T=\frac{\sqrt{x+5}}{x-1} $$**  
- The square root $$ \sqrt{x+5} $$ is defined for $$ x+5 \ge 0 $$, which gives  
  $$
  x \ge -5.
  $$  
- The denominator $$ x-1 $$ must not be zero, hence  
  $$
  x \ne 1.
  $$

**Step 2. Analyze each candidate value $$ x \in \{-6, -2, 1, 4\} $$**
- For $$ x=-6 $$:  
  - Check the square root: $$ -6+5=-1 $$, so  
    $$
    \sqrt{-1} \quad \text{is non-real.}
    $$
    
- For $$ x=-2 $$:  
  - Check the square root: $$ -2+5=3 $$, so  
    $$
    \sqrt{3} \quad \text{is real but irrational.}
    $$
    
- For $$ x=1 $$:  
  - The denominator is $$ 1-1=0 $$, therefore $$ T $$ is undefined.
    
- For $$ x=4 $$:  
  - Check the square root: $$ 4+5=9 $$, so  
    $$
    \sqrt{9}=3.
    $$
  - The denominator is $$ 4-1=3 $$.  
  - Thus,  
    $$
    T=\frac{3}{3}=1, \quad \text{which is a rational number.}
    $$

**Step 3. Conclusion**  
- The value of $$ x $$ for which $$ T $$ is rational is  
  $$
  x=4.
  $$
- The value of $$ x $$ for which $$ T $$ is non-real is  
  $$
  x=-6.
  $$
For $$ x = 4 $$, $$ T $$ is rational. For $$ x = -6 $$, $$ T $$ is non-real.

ESTION 1 Given: \( T=\frac{\sqrt{x+5}}{x-1} \), where \( x \in\{-6 ;-2 ; 1 ; 4\} \). Choose from the given set of numbers a value of \( x \) for which \( T \) will be: 1.I.1 rational 1.1.2 non-real

Upstudy AI Solution

Answer

Solution

Bonus Knowledge

Related Questions

Latest Algebra Questions