rac{2x - 1}{x + 3} - 1 > 0 \Rightarrow rac{2x - 1 - (x + 3)}{x + 3} > 0 \Rightarrow rac{2x - 1 - x - 3}{x + 3} > 0 \Rightarrow rac{x - 4}{x + 3} > 0
["Solving the Inequality RAC{2x − 1}{x + 3} − 1 > 0: A Step-by-Step Guide", "When solving rational inequalities like RAC{2x − 1}{x + 3} − 1 > 0, a key strategy is transforming the expression into a simpler, solvable form. This article walks you through the step-by-step process of solving RAC{2x − 1}{x + 3} − 1 > 0, explaining each transformation and revealing how the inequality can be analyzed efficiently.", "---", "### Understanding the Inequality", "We start with:", "[\n\frac{2x - 1}{x + 3} - 1 > 0\n]", "This inequality asks for the values of ( x ) where the expression on the left is positive. To solve it, we simplify the expression by combining the terms into a single rational function.", "---", "### Step 1: Subtract 1 and Combine the Expression", "We rewrite 1 as a fraction with the same denominator:", "[\n\frac{2x - 1}{x + 3} - \frac{(x + 3)}{x + 3} > 0\n]", "Now subtract the numerators:", "[\n\frac{(2x - 1) - (x + 3)}{x + 3} > 0\n]", "Simplify the numerator:", "[\n2x - 1 - x - 3 = (2x - x) + (-1 - 3) = x - 4\n]", "So the inequality becomes:", "[\n\frac{x - 4}{x + 3} > 0\n]", "---", "### Step 2: Analyzing the Rational Inequality", "We now solve:", "[\n\frac{x - 4}{x + 3} > 0\n]", "This is a rational inequality, and to find where it's positive, we identify:", "- The critical points: values of ( x ) that make the numerator or denominator zero.", "Set numerator and denominator to zero:", "- ( x - 4 = 0 \Rightarrow x = 4 )\n- ( x + 3 = 0 \Rightarrow x = -3 )", "These divide the real number line into intervals:", "1. ( (-\infty, -3) )\n2. ( (-3, 4) )\n3. ( (4, \infty) )", "We test a point in each interval to determine the sign of the expression.", "---", "### Step 3: Test Sign of the Expression", "- In (–∞, –3): Try ( x = -4 )", "[\n \frac{-4 - 4}{-4 + 3} = \frac{-8}{-1} = + \quad \ ext{(Positive)}\n ]", "- In (–3, 4): Try ( x = 0 )", "[\n \frac{0 - 4}{0 + 3} = \frac{-4}{3} = - \quad \ ext{(Negative)}\n ]", "- In (4, ∞): Try ( x = 5 )", "[\n \frac{5 - 4}{5 + 3} = \frac{1}{8} = + \quad \ ext{(Positive)}\n ]", "The expression is positive in intervals (–∞, –3) and (4, ∞).", "---", "### Step 4: Final Solution", "Since the inequality is strict (( > 0 )), we exclude points where the expression equals zero or is undefined.", "- The expression is zero at ( x = 4 ), so we exclude this point.\n- The expression is undefined at ( x = -3 ), so this value is also excluded.", "Thus, the solution is:", "[\nx \in (-\infty, -3) \cup (4, \infty)\n]", "---", "### Summary", "Solving RAC{2x − 1}{x + 3} − 1 > 0 reduces elegantly to a single rational expression:", "1. Combine terms:\n [\n \frac{x - 4}{x + 3} > 0\n ]\n2. Identify critical points: ( x = -3 ) and ( x = 4 ).\n3. Test intervals to determine where the quotient is positive.\n4. Exclude undefined and zero points for strict inequality.", "Final answer:\n[\n\boxed{x \in (-\infty, -3) \cup (4, \infty)}\n]", "---", "Why Is This Important?\nMastering rational inequalities through step-by-step simplification helps solve real-world problems in economics, physics, and engineering. Understanding sign analysis and domain restrictions strengthens problem-solving skills for advanced mathematics.", "---", "Keywords for SEO:\n[\n\frac{2x - 1}{x + 3} - 1 > 0,\quad RAC{2x - 1}{x + 3} - 1 > 0,\quad \frac{x - 4}{x + 3} > 0,\quad rational inequality solution, solving rational inequalities step-by-step\n]"]
