Determine the largest integer \( x \) that satisfies the inequality \( 2x^2 - 9x + 7 < 0 \).
["# Determine the Largest Integer ( x ) That Satisfies the Inequality ( 2x^2 - 9x + 7 < 0 )", "Understanding quadratic inequalities is essential in algebra, and one common challenge is finding the largest integer solution to expressions like ( 2x^2 - 9x + 7 < 0 ). In this article, we’ll walk through the steps to solve this inequality and identify the largest integer ( x ) that satisfies it.", "## Step 1: Analyze the Quadratic Expression", "We start with the inequality:", "[\n2x^2 - 9x + 7 < 0\n]", "This is a quadratic inequality in standard form ( ax^2 + bx + c < 0 ), where ( a = 2 ), ( b = -9 ), and ( c = 7 ). Since the coefficient of ( x^2 ) is positive (( a = 2 > 0 )), the parabola opens upwards. Therefore, the expression ( 2x^2 - 9x + 7 ) is negative between the two real roots.", "## Step 2: Find the Roots of the Corresponding Equation", "To find where the quadratic equals zero, solve:", "[\n2x^2 - 9x + 7 = 0\n]", "Use the quadratic formula:", "[\nx = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\n]", "Substitute ( a = 2 ), ( b = -9 ), and ( c = 7 ):", "[\nx = \frac{9 \pm \sqrt{(-9)^2 - 4(2)(7)}}{2(2)} = \frac{9 \pm \sqrt{81 - 56}}{4} = \frac{9 \pm \sqrt{25}}{4} = \frac{9 \pm 5}{4}\n]", "Compute both roots:", "[\nx = \frac{9 + 5}{4} = \frac{14}{4} = 3.5 \quad \ ext{and} \quad x = \frac{9 - 5}{4} = \frac{4}{4} = 1\n]", "So the roots are ( x = 1 ) and ( x = 3.5 ).", "## Step 3: Determine the Interval Where the Inequality Holds", "Since the parabola opens upward, the inequality ( 2x^2 - 9x + 7 < 0 ) holds strictly between the roots:", "[\n1 < x < 3.5\n]", "This means ( x ) must be greater than 1 and less than 3.5.", "## Step 4: Identify the Largest Integer in the Interval", "We are looking for the largest integer value of ( x ) such that ( 1 < x < 3.5 ). The integers in this range are:", "[\nx = 2 \quad \ ext{and} \quad x = 3\n]", "Since 3 is less than 3.5 and greater than 1, the largest integer satisfying the inequality is:", "[\n\boxed{3}\n]", "## Step 5: Verification", "Check ( x = 3 ):", "[\n2(3)^2 - 9(3) + 7 = 18 - 27 + 7 = -2 < 0 \quad \ ext{✅}\n]", "Check ( x = 4 ):", "[\n2(4)^2 - 9(4) + 7 = 32 - 36 + 7 = 3 > 0 \quad \ ext{✖ (not valid)}\n]", "Thus, ( x = 3 ) is indeed the largest integer solution.", "---", "Conclusion:\nThe largest integer ( x ) satisfying ( 2x^2 - 9x + 7 < 0 ) is ( \boxed{3} ). Understanding the sign of quadratic expressions between roots helps efficiently solve such inequalities—key knowledge for students and math enthusiasts alike."]
