- The given equation is 32x+3 – 244(3x) = – 9. - This can be rewritten using exponent rules: 32x 33 – 244 3x = – 9. - This simplifies to 27 (3x)2 – 244 3x + 9 = 0. - Let y = 3x. Since 3x is always positive for real x, y must be a positive value. - Substituting y into the equation gives a quadratic equation in y: 27y2 – 244y + 9 = 0. - We can solve this quadratic equation for y using the quadratic formula $y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$: $y = \frac{244 \pm \sqrt{(-244)^2 - 4(27)(9)}}{2(27)}$ $y = \frac{244 \pm \sqrt{59536 - 972}}{54}$ $y = \frac{244 \pm \sqrt{58564}}{54}$ $y = \frac{244 \pm 242}{54}$ - This gives two possible values for y: $y_1 = \frac{244 + 242}{54} = \frac{486}{54} = 9$ $y_2 = \frac{244 - 242}{54} = \frac{2}{54} = \frac{1}{27}$ - Both values of y (9 and 1/27) are positive, so they are valid solutions for y = 3x. - Now, we find the corresponding values of x by substituting back y = 3x: - For y1 = 9: 3x = 9 => 3x = 32 => x = 2. This is a positive number. - For y2 = 1/27: 3x = 1/27 => 3x = 3-3 => x = -3. This is a negative number. - The solutions for x are 2 and -3. - The statement "x is a positive number" is true for x=2 but false for x=-3. - The statement "x is a negative number" is false for x=2 but true for x=-3. - The statement "x can be either positive or negative number" is true because one solution (x=2) is positive and the other solution (x=-3) is negative.