Fractional Quadratic Inequality

[mathdad]

Solve the inequality: (3x+1)/(4x-2) is less than or equal to 5.

Solution:

(3x + 1)/(4x – 2) ≤ 5

Multiply both sides by the given denominator.

3x + 1 ≤ 5(4x –2)

Distribute 5 on the right side.

3x + 1 ≤ 20x –10

Simplify

11 ≤ 17x

11/17 ≤ x

or

x ≥ 11/17

Correct?

Note: I think [ or ] not [ and ] applies in the answer. Yes?

 

[Romsek]

You neglect the case where \(\displaystyle 4x-2<0\)

In this case the inequality sign would flip due to multiplying by a negative number.

 

[pka]

Solve the inequality: (3x+1)/(4x-2) is less than or equal to 5.
Solution:
(3x + 1)/(4x – 2) ≤ 5
Multiply both sides by the given denominator. NO!

That is a bad bad idea. At first we do not know whether it is positive or negative..
Do this:
\(\displaystyle \begin{align*}\frac{3x+1}{4x-2}&\le 5 \\\frac{3x+1}{4x-2}-5&\le 0\\\frac{3x+1}{4x-2}-\frac{5(4x-2)}{4x-2}&\le 0 \end{align*}\)
Now expand, combine into one fraction and finish.

 

[Steven G]

Unless you are going to amazingly careful NEVER multiply both sides of an inequality by an expression that is not always positive.

 

[mathdad]

Thank you everyone.