How to solve (6x)/(x-1) <= 1

Q: How to solve (6x)/(x-1) <= 1

1. Factor the equation (x + 2)(x + 3) = 0. 2. Set each factor equal to zero: x + 2 = 0 and x + 3 = 0. 3. Solve for x: x = -2 and x = -3.

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Question

$$\frac{ 6x }{ x-1 } \leq 1$$

Answer

-1/5<=x<1

Solution

Replace the inequality sign with an equal sign, so that we can solve it like a normal equation.

\[\frac{6x}{x-1}=1\]

Multiply both sides by $x-1$.

\[6x=x-1\]

Subtract $x$ from both sides.

\[6x-x=-1\]

Simplify $6x-x$ to $5x$.

\[5x=-1\]

Divide both sides by $5$.

\[x=-\frac{1}{5}\]

Also note that $x$ is undefined at $1$.

\[x\ne 1\]

From the values of $x$ above, we have these 3 intervals to test.

\[\begin{aligned}&x\le -\frac{1}{5}\\&-\frac{1}{5}\le x\le 1\\&x\ge 1\end{aligned}\]

Pick a test point for each interval.

For the interval $x\le -\frac{1}{5}$:

Let's pick $x=-1$. Then, $\frac{6\times -1}{-1-1}\le 1$.After simplifying, we get $3\le 1$, which is

false

Drop this interval.

For the interval $-\frac{1}{5}\le x\le 1$:

Let's pick $x=0$. Then, $\frac{6\times 0}{0-1}\le 1$.After simplifying, we get $0\le 1$, which is

true

Keep this interval.

For the interval $x\ge 1$:

Let's pick $x=2$. Then, $\frac{6\times 2}{2-1}\le 1$.After simplifying, we get $12\le 1$, which is