91Ó°ÊÓ

Graphing solution sets for a rational inequality can initially seem daunting, but it's just about understanding where the inequality holds true. The goal is to find all the values of the variable that make the inequality correct. In this case, the inequality \[ \frac{3x}{x-1} \leq \frac{x}{x+4} + 3 \] was simplified and analyzed, leading us to discover critical points:

• The numerator of the simplified inequality \(2x^2 + 3x - 12 = 0\) gives us \(x = 2\) and \(x = -3\). These points could change the sign of the expression, hence creating potential solution boundaries.
• The denominator is undefined at \(x = 1\) and \(x = -4\), so these points divide the number line into different regions to test.

To graph the solution set, plot a number line and mark these critical points. The goal is to determine in which intervals or regions the inequality holds true. Here, after testing, the solution set is the interval \((-\infty, -4)\), where the inequality is valid. Remember, just shade the line that represents the actual solution, ensuring to exclude points where the expression is undefined or does not satisfy the inequality.

Finding a common denominator is essential when dealing with rational inequalities that involve sums of fractions. In these scenarios, you need to simplify the expression to a standard form, allowing for easy comparison and manipulation across the equation. In our exercise, the fractions \(\frac{3x}{x-1} \) and \(\frac{x}{x+4}\) were present on the left side of the inequality. A common denominator \((x-1)(x+4)\) was identified to facilitate the combination of these terms.

Once you establish and use a common denominator, you can rewrite the expression as a single fraction, making further analysis and solving more streamlined. The common denominator also helps in identifying undefined regions for the inequality, since any value that would make the denominator zero must be excluded from the solutions. Remember, matching denominators is a crucial step to ensure coherent results when working with rational inequalities.

Examining the numerator and denominator separately is a key step in solving rational inequalities, as each component contributes to the behavior of the expressed inequality. For our simplified inequality, \[ \frac{2x^2+3x-12}{(x-1)(x+4)} \leq 0 \], analyzing where the numerator becomes zero helps us identify major turning points in the inequality's behavior.

• Resolve the equation \(2x^2 + 3x - 12 = 0\) to find values that nullify the numerator, obtaining the critical points \(x = 2\) and \(x = -3\).
• Similarly, determining where the denominator becomes zero, \((x-1)(x+4) eq 0\), indicates where the function is undefined and potentially changes the inequality's sign.

Together, these analyses help split the number line into distinct regions that can either satisfy or disprove the initial inequality dependent on the sign of the resulting expression. By conducting a comprehensive analysis of both numerator and denominator, you can systematically test each segment to determine viable solution areas. Critical points should be used as boundaries for testing different intervals on the number line, ensuring a precise solution set.