MCQ
Solve the following in equations $\frac{2\text{x}+4}{\text{x}-1}\geq5$
- A$\big(1,2\big]$
- B$\big(1,3\big)$
- ✓$\big(1,3\big]$
- D$\big(1,4\big]$
✓
Answer
Correct option: C.
$\big(1,3\big]$
$\frac{2\text{x}+4}{\text{x}-1}\geq5$
$\Rightarrow\frac{2\text{x}+4}{\text{x}-1}-5\geq0$
$\Rightarrow\frac{2\text{x}+4-5(\text{x-1})}{\text{x-1}}\geq0$
$\Rightarrow\frac{2\text{x}+4-\text{5x+5}}{\text{x-1}}\geq0$
$\Rightarrow\frac{9-3\text{x}}{\text{x-1}}\geq0$
$\Rightarrow\frac{-3\text{(x-3)}}{\text{x}-1}\geq0$
Multiplying each side of an inequality by a negative number $\big(\frac{-1}{3}\big)$ reverses the direction of the inequality symbol.
$\Rightarrow\frac{(\text{x}-3)}{\text{x}-1}\leq0$ Here $\text{x}-1\neq0$
$\Rightarrow\text{x}\neq1$
Hence the solution set of the given in equations is $\big(1, 3\big]$
$\Rightarrow\frac{2\text{x}+4}{\text{x}-1}-5\geq0$
$\Rightarrow\frac{2\text{x}+4-5(\text{x-1})}{\text{x-1}}\geq0$
$\Rightarrow\frac{2\text{x}+4-\text{5x+5}}{\text{x-1}}\geq0$
$\Rightarrow\frac{9-3\text{x}}{\text{x-1}}\geq0$
$\Rightarrow\frac{-3\text{(x-3)}}{\text{x}-1}\geq0$
Multiplying each side of an inequality by a negative number $\big(\frac{-1}{3}\big)$ reverses the direction of the inequality symbol.
$\Rightarrow\frac{(\text{x}-3)}{\text{x}-1}\leq0$ Here $\text{x}-1\neq0$
$\Rightarrow\text{x}\neq1$
Hence the solution set of the given in equations is $\big(1, 3\big]$
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