Question

what is the value of ( x ) in the inequality: (\frac{2 - 5x}{-3}+4 < -x)
( \bigcirc x > -\frac{5}{4} )
( \bigcirc x < \frac{4}{5} )
( \bigcirc x < -\frac{5}{4} )
( \bigcirc x < \frac{5}{4} )

Explanation:

Step1: Simplify the left - hand side fraction

We have the inequality \(\frac{2 - 5x}{-3}+4\lt - x\). First, simplify \(\frac{2 - 5x}{-3}\), which is equal to \(\frac{-(5x - 2)}{-3}=\frac{5x - 2}{3}\) (or we can also think of it as multiplying the numerator and denominator by - 1, and when we do that, the sign of the fraction changes). So the inequality becomes \(\frac{5x - 2}{3}+4\lt - x\).

Step2: Eliminate the fraction

Multiply each term in the inequality by 3 to get rid of the denominator. We have \(3\times\frac{5x - 2}{3}+3\times4\lt3\times(-x)\). This simplifies to \(5x - 2 + 12\lt - 3x\).

Step3: Combine like terms

Simplify the left - hand side: \(5x+10\lt - 3x\).

Step4: Move all x terms to one side

Add \(3x\) to both sides of the inequality: \(5x + 3x+10\lt - 3x+3x\), which gives \(8x + 10\lt0\). Then subtract 10 from both sides: \(8x+10 - 10\lt0 - 10\), so \(8x\lt - 10\). Wait, we made a mistake in step 1. Let's go back.

Correct Step 1: The original inequality is \(\frac{2 - 5x}{-3}+4\lt - x\). When we have \(\frac{2 - 5x}{-3}\), we can rewrite it as \(\frac{-(5x - 2)}{-3}=\frac{5x - 2}{3}\) is wrong. Actually, \(\frac{a}{-b}=-\frac{a}{b}\), so \(\frac{2 - 5x}{-3}=-\frac{2 - 5x}{3}=\frac{5x - 2}{3}\) is correct, but let's do it another way. Let's multiply both sides of the inequality by - 3. But we have to remember that when we multiply or divide an inequality by a negative number, the direction of the inequality sign changes.

So, multiply both sides of \(\frac{2 - 5x}{-3}+4\lt - x\) by - 3:

\((\frac{2 - 5x}{-3})\times(-3)+4\times(-3)\gt(-x)\times(-3)\) (because we multiplied by a negative number, the \(\lt\) becomes \(\gt\))

This simplifies to \(2 - 5x-12\gt3x\)

Step 2: Combine like terms: \(2-12 - 5x\gt3x\), so \(-10 - 5x\gt3x\)

Step 3: Add \(5x\) to both sides: \(-10-5x + 5x\gt3x + 5x\), which gives \(-10\gt8x\)

Step 4: Divide both sides by 8: \(\frac{-10}{8}\gt x\), simplify \(\frac{-10}{8}=-\frac{5}{4}\), so \(x\lt-\frac{5}{4}\)

Answer:

\(x\lt-\frac{5}{4}\) (corresponding to the option \(x\lt-\frac{5}{4}\))