Verified Answer :

  • Start by factoring out the constant 3 from the quadratic expression: \(3(x^2 + x - 2) \leq 0\).
  • Factor the inner quadratic expression: \(3(x + 2)(x - 1) \leq 0\).
  • Identify the roots of the quadratic equation, which are \(x = -2\) and \(x = 1\).
  • Determine that the solution for the inequality is the interval between the roots, inclusive: \(\boxed{-2 \leq x \leq 1}\).

Explanation

  1. Problem Analysis and Strategy

We are tasked with solving the quadratic inequality \(3x^2 + 3x - 6 \leq 0\). A quadratic inequality of the form \(ax^2 + bx + c \leq 0\) can be solved by identifying the roots of the corresponding quadratic equation \(ax^2 + bx + c = 0\) and then analyzing the intervals determined by these roots. Since the leading coefficient is positive, the graph of the function \(f(x) = 3x^2 + 3x - 6\) is a parabola that opens upwards, meaning the function takes on values less than or equal to zero between its two roots.

  1. Factoring the Quadratic Expression

First, we simplify the inequality by factoring out the common factor of 3:
\[3(x^2 + x - 2) \leq 0\]
Next, we factor the quadratic expression \(x^2 + x - 2\). We look for two numbers that multiply to -2 and add up to 1. These numbers are 2 and -1. Thus, we can write:
\[x^2 + x - 2 = (x + 2)(x - 1)\]
Substituting this back into our inequality, we have:
\[3(x + 2)(x - 1) \leq 0\]
Dividing both sides by 3 gives:
\[(x + 2)(x - 1) \leq 0\]

  1. Identifying the Solution Interval

To solve \((x + 2)(x - 1) \leq 0\), we determine the roots of the equation \((x + 2)(x - 1) = 0\), which are \(x = -2\) and \(x = 1\). These roots divide the number line into three intervals: \(x < -2\), \(-2 \leq x \leq 1\), and \(x > 1\). Testing a value in each interval or using the property that a upward-opening parabola is non-positive between its roots, we find that the inequality holds when \(x\) is between the roots, inclusive. Therefore, the solution set is \([-2, 1]\).

  1. Final Conclusion

Comparing our result \([-2, 1]\) with the provided options, we see that it corresponds to the second option \(-2 \leq x \leq 1\). The smallest value is -2 and the largest is 1, inclusive. Thus, the correct solution is \(\boxed{-2 \leq x \leq 1}\).

Examples

Quadratic inequalities are frequently used in engineering and physics to determine the safe range for a variable. For example, if the height of a projectile relative to a target is given by a quadratic expression, finding the interval where the height is less than or equal to a certain value helps in calculating the safe landing zone or the range of impact. By factoring the quadratic expression, one can quickly identify the boundaries for safety and make precise adjustments.

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