Question

1. choose the correct graph of the rational inequality with the correct restriction line.

$y < \frac{x^2 - 5x + 4}{x - 4}$

Explanation:

Step1: Simplify the Rational Expression

First, factor the numerator \(x^{2}-5x + 4\). We know that \(x^{2}-5x + 4=(x - 1)(x - 4)\). So the inequality becomes \(y<\frac{(x - 1)(x - 4)}{x - 4}\), with the restriction \(x
eq4\) (since the denominator cannot be zero). After canceling out \((x - 4)\) (for \(x
eq4\)), we get \(yeq4\)).

Step2: Analyze the Graph Features

The simplified inequality is a linear inequality \(yeq4\)). We need to check which graph has a dashed line for \(y=x - 1\) (slope \(1\), y - intercept \(-1\)) and a vertical dashed line at \(x = 4\), and the region below the line \(y=x - 1\) (excluding the line and the point at \(x = 4\)). The first graph (top one) shows a dashed line with slope \(1\), vertical dashed line at \(x = 4\), and the shaded region below the line, which matches our analysis. The second graph has a different slope and doesn't match the simplified line \(y=x - 1\).

Answer:

The top graph (the first one with the grid and dashed lines, including the vertical dashed line at \(x = 4\) and the dashed line \(y=x - 1\) with the region below shaded)