Question

1. the positive non - permissible value for the expression \\(dfrac{(3x + 2)(2x - 5)}{(4x - 1)(5x + 2)}), correct to the nearest hundredth, is ______.

Explanation:

Step1: Identify Denominator Factors

The expression's denominator is \((4x - 1)(5x + 2)\). Non - permissible values occur when the denominator is zero. So we set each factor equal to zero:

• For \(4x-1 = 0\), solve for \(x\): \(4x=1\), so \(x=\frac{1}{4}=0.25\).
• For \(5x + 2=0\), solve for \(x\): \(5x=-2\), so \(x =-\frac{2}{5}=-0.4\).

We also need to check if there are any non - permissible values from the numerator? No, because non - permissible values for a rational expression come from the denominator being zero (since division by zero is undefined). But wait, let's re - examine the original expression. Wait, the expression is \(\frac{(3x + 2)(2x-5)}{(4x - 1)(5x + 2)}\). So we set the denominator equal to zero:
\(4x-1 = 0\) gives \(x=\frac{1}{4}=0.25\)
\(5x + 2=0\) gives \(x=-\frac{2}{5}=-0.4\)

We want the positive non - permissible value. Among \(0.25\) and \(- 0.4\), the positive one is \(0.25\). Wait, but let's check again. Wait, maybe I made a mistake. Wait, the denominator is \((4x - 1)(5x+2)\). So when \(4x - 1=0\), \(x = 1/4=0.25\); when \(5x + 2=0\), \(x=-2/5=-0.4\). The positive non - permissible value is \(0.25\). But wait, let's confirm.

Wait, is there any other factor? No, the denominator is \((4x - 1)(5x + 2)\). So the non - permissible values are \(x = 1/4\) and \(x=-2/5\). The positive one is \(1/4 = 0.25\).

Step2: Determine Positive Non - Permissible Value

We have two non - permissible values \(x = 0.25\) (positive) and \(x=-0.4\) (negative). So the positive non - permissible value is \(0.25\).

Answer:

\(0.25\)