Final answer:
To find the number of real solutions to an equation like -11x² - x - 11 = 0, one must rearrange it into standard quadratic form and apply the quadratic formula. The number of real solutions depends on the discriminant (b² - 4ac) which determines if the solutions are real and distinct, one real, or complex.
Step-by-step explanation:
The question asks about the number of real solutions to the equation –211x² = x + 11. To find the real solutions, we need to rearrange the equation into a quadratic equation form ax² + bx + c = 0 and then apply the quadratic formula or other solution methods as appropriate. In this case, the equation simplifies to -11x² - x - 11 = 0. Applying the quadratic formula, x = ∛[b² - 4ac]/(2a), will yield two solutions for x, which can be real or complex. If the discriminant (b² - 4ac) is positive, there will be two distinct real solutions; if it is zero, there will be exactly one real solution; if it is negative, there will be no real solutions, only complex solutions.
Significance: In certain contexts, it might be the case that only one of the real solutions is meaningful or reasonable, as with the example provided regarding the typical freeway on-ramp time of 10 seconds. In other contexts, both real solutions can make sense and be applied to the problem at hand.