When \( y < -1 \), \( \frac{y+1}{y-1} > 0 \), e.g., \( y = -2 \Rightarrow x^2 = \frac{-1}{-3} = \frac{1}{3} \), real \( x \).

["# When ( y < -1 ), Why ( \frac{y+1}{y-1} > 0 ) — A Clear Explanation with Example", "Understanding inequalities involving rational expressions is essential in algebra, especially when determining intervals where expressions stay positive or negative. One common question studied in precalculus and algebra is:\nWhen ( y < -1 ), is ( \frac{y+1}{y-1} > 0 )?\nIn this article, we’ll explore this inequality step-by-step, explain its logic using real-world examples, and demonstrate how real solutions exist in this domain—like when ( y = -2 ) leads to a real ( x ) satisfying ( x^2 = \frac{-1}{-3} = \frac{1}{3} ).", "---", "## The Inequality: ( \frac{y+1}{y-1} > 0 )", "To analyze when this rational expression is positive, we examine the signs of the numerator and the denominator.", "The expression is:\n[\n\frac{y+1}{y-1} > 0\n]", "This inequality holds true when both the numerator and denominator are positive, or when both are negative.", "### Step 1: Find critical points\nThe expression is undefined when the denominator is zero:\n[\ny - 1 = 0 \Rightarrow y = 1\n]\nThe numerator is zero when:\n[\ny + 1 = 0 \Rightarrow y = -1\n]\nSo, critical points divide the number line:\n[\ny = -1,\quad y = 1\n]", "These split the real line into intervals:\n1. ( (-\infty, -1) )\n2. ( (-1, 1) )\n3. ( (1, \infty) )", "---", "## Step 2: Analyze sign changes across intervals", "| Interval | Test value | ( y+1 ) | ( y-1 ) | ( \frac{y+1}{y-1} ) | Positive? |\n|----------------|-----------|----------|---------|----------------------|-----------|\n| ( (-\infty, -1) ) | ( y = -2 ) | ( -1 ) | ( -3 ) | ( \frac{-1}{-3} = \frac{1}{3} > 0 ) | Yes |\n| ( (-1, 1) ) | ( y = 0 ) | ( +1 ) | ( -1 ) | ( \frac{+1}{-1} = -1 < 0 ) | No |\n| ( (1, \infty) ) | ( y = 2 ) | ( +3 ) | ( +1 ) | ( \frac{3}{1} = 3 > 0 ) | Yes |", "### Summary:\n- ( \frac{y+1}{y-1} > 0 ) when ( y < -1 ) or ( y > 1 )\n- The inequality is never true on ( (-1, 1) ), including ( y = -1 ) (numerator zero, expression = 0), and undefined at ( y = 1 )", "---", "## Visual Insight: Example with ( y = -2 )", "Let’s take ( y = -2 ), satisfying ( y < -1 ). Computationally:", "[\n\frac{y+1}{y-1} = \frac{-2 + 1}{-2 - 1} = \frac{-1}{-3} = \frac{1}{3} > 0\n]\nThis confirms the inequality holds. Moreover, solving ( x^2 = \frac{y+1}{y-1} = \frac{1}{3} ) yields real solutions:\n[\nx = \pm \sqrt{\frac{1}{3}} = \pm \frac{\sqrt{3}}{3}\n]\nThis illustrates: when ( y < -1 ), the expression becomes a positive real number, and its square root is real.", "---", "## Why Does This Matter?", "Understanding sign behavior of rational functions helps:\n- Solve inequalities in algebra and calculus\n- Analyze function behavior and continuity\n- Apply to real-world modeling (e.g., profit margins, motion equations)", "The interval ( y < -1 ) produces a positive rational value, which guarantees real solutions when appearing under a square root or inside square roots—critical for real number results.", "---", "## Final Thoughts", "When ( y < -1 ),\n[\n\frac{y+1}{y-1} > 0\n]\nis indeed true. This rational expression is positive in the domain ( y \in (-\infty, -1) \cup (1, \infty) ), and especially useful when yielding positive values needed for real ( x ) in equations like ( x^2 = \frac{y+1}{y-1} ). Using examples such as ( y = -2 ) confirms both the inequality and the feasibility of real roots—making these concepts concrete and powerful tools in mathematics.", "---", "Key Takeaways:**\n- The inequality ( \frac{y+1}{y-1} > 0 ) holds if ( y < -1 ) or ( y > 1 )\n- For ( y < -1 ), the expression is positive and real\n- Real solutions exist for ( x ) when ( x^2 = \frac{y+1}{y-1} > 0 ), such as ( x = \pm \sqrt{\frac{1}{3}} ) when ( y = -2 )\n- Mastering sign analysis unlocks deeper algebra and calculus insights", "Happy calculating—especially when inequalities align neatly for real solutions!"]