Questions: Inequality solution (2x-1)(3x-1) ≤ 7 x=

Solution Steps

To solve the inequality \((2x-1)(3x-1) \leq 7\), we first need to expand the left-hand side and then bring all terms to one side of the inequality to form a quadratic inequality. Next, we solve the corresponding quadratic equation to find the critical points. Finally, we test intervals determined by these critical points to find the solution set for the inequality.

We start with the inequality: \[ (2x - 1)(3x - 1) \leq 7 \] Expanding the left-hand side gives: \[ 6x^2 - 5x - 1 \leq 7 \]

Next, we rearrange the inequality to bring all terms to one side: \[ 6x^2 - 5x - 8 \leq 0 \]

To solve the quadratic inequality \(6x^2 - 5x - 8 \leq 0\), we find the roots of the corresponding equation: \[ 6x^2 - 5x - 8 = 0 \] The roots are: \[ x = -\frac{2}{3} \quad \text{and} \quad x = \frac{3}{2} \]

The quadratic opens upwards (since the coefficient of \(x^2\) is positive), so the solution to the inequality \(6x^2 - 5x - 8 \leq 0\) is between the roots: \[ -\frac{2}{3} \leq x \leq \frac{3}{2} \]

Final Answer