Solve the quadratic inequality \( 2x^2 - 9x + 7 < 0 \).

["# Solve the Quadratic Inequality ( 2x^2 - 9x + 7 < 0 ): Step-by-Step Guide", "Solving quadratic inequalities is a fundamental skill in algebra that helps determine where expressions are positive, negative, or zero. In this article, we’ll walk you through solving the inequality:\n$$\n2x^2 - 9x + 7 < 0\n$$\nWe’ll explore how to find the roots, analyze the parabola's shape, and determine the solution set clearly and thoroughly.", "---", "## Step 1: Understand the Inequality", "We are solving:\n$$\n2x^2 - 9x + 7 < 0\n$$\nThis means we want to find all real values of ( x ) for which the quadratic expression is less than zero, i.e., negative.", "Since the coefficient of ( x^2 ) is positive (( a = 2 > 0 )), the parabola opens upward and thus lies below the x-axis only between its two real roots.", "---", "## Step 2: Find the Roots Using the Quadratic Formula", "We solve the corresponding equation:\n$$\n2x^2 - 9x + 7 = 0\n$$\nUsing the quadratic formula:\n$$\nx = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\n$$\nwhere ( a = 2 ), ( b = -9 ), ( c = 7 ).", "Calculate the discriminant:\n$$\n\Delta = (-9)^2 - 4(2)(7) = 81 - 56 = 25\n$$\nSince ( \Delta = 25 > 0 ), there are two distinct real roots.", "Now compute the roots:\n$$\nx = \frac{9 \pm \sqrt{25}}{4} = \frac{9 \pm 5}{4}\n$$\nSo,\n$$\nx_1 = \frac{9 - 5}{4} = \frac{4}{4} = 1\n$$\n$$\nx_2 = \frac{9 + 5}{4} = \frac{14}{4} = \frac{7}{2} = 3.5\n$$", "---", "## Step 3: Analyze the Inequality Using the Roots", "Since the parabola opens upward and the expression is less than zero, the inequality ( 2x^2 - 9x + 7 < 0 ) holds between the two roots. That is:", "$$\n1 < x < 3.5\n$$", "At ( x = 1 ) and ( x = 3.5 ), the expression equals zero, but the inequality is strict, so these endpoints are not included.", "---", "## Step 4: Write the Final Solution", "The solution to the inequality ( 2x^2 - 9x + 7 < 0 ) is all real numbers ( x ) such that:", "$$\n\boxed{1 < x < \frac{7}{2}}\n$$", "---", "## Summary", "- The quadratic opens upward (( a > 0 )), so it’s negative between its roots.\n- Solved ( 2x^2 - 9x + 7 = 0 ) using the quadratic formula to find roots at ( x = 1 ) and ( x = 3.5 ).\n- The solution is the open interval ( (1, 3.5) ).", "---", "## Key Takeaways", "- When solving ( ax^2 + bx + c < 0 ) and ( a > 0 ), the solution is the interval between the roots.\n- Always verify by testing values in each interval.\n- Graphing the function confirms the behavior: downward between roots where the parabola dips below the x-axis.", "---", "## Frequently Asked Questions (FAQ)", "Q: Why isn’t the inequality greater than zero here?\nA: Because the parabola opens upward, so it is negative between the roots, not outside.", "Q: Can I graph this quickly to check?\nA: Yes! Plotting ( y = 2x^2 - 9x + 7 ) shows a U-shaped curve intersecting the x-axis at 1 and 3.5, confirming the solution is where the graph is below the x-axis — between these points.", "Q: What if the roots were equal or imaginary?\nA: If roots were repeated (Δ = 0), the parabola just touches the x-axis — solution set excludes the root if the inequality is strict. For imaginary roots (Δ < 0), the quadratic never crosses the x-axis; if ( a > 0 ), it’s always positive, so no solution.", "---", "Mastering quadratic inequalities opens doors to more advanced math — keep practicing!\nFor more guides, visit our algebra help center.", "---", "Keywords: solve quadratic inequality, 2x² - 9x + 7 < 0, step-by-step solution, quadratic inequality tutorial, algebra help"]