Verified Answer :

  • The inequality \((x-1)(x+2)(2x-7)^2 \leq 0\) is analyzed by considering the factors.
  • The squared term \((2x-7)^2\) is always non-negative and equals 0 at \(x = 3.5\), which is a solution.
  • For \(x eq 3.5\), the inequality simplifies to \((x-1)(x+2) \leq 0\), which yields the interval \([-2, 1]\).
  • The complete solution set is the union of the interval and the isolated root: \(\boxed{[-2, 1] \cup \{3.5\}}\).

Explanation

  1. Analysis of the Inequality

We are given the inequality \((x-1)(x+2)(2x-7)^2 \leq 0\). First, note that the expression is a product of three factors: \((x-1)\), \((x+2)\), and \((2x-7)^2\). We observe that the term \((2x-7)^2\) is the square of a real number, meaning it is non-negative for all real values of \(x\), i.e., \((2x-7)^2 \geq 0\). The expression equals zero when any of the factors is zero. This happens at \(x = 1\), \(x = -2\), and \(x = \frac{7}{2}\).

  1. Splitting into Cases

Since \((2x-7)^2 \geq 0\) for all real \(x\), we can distinguish two scenarios.
Scenario 1: \((2x-7)^2 = 0\). This happens when \(x = \frac{7}{2} = 3.5\). At this point, the entire product is zero, satisfying the inequality \(\leq 0\).
Scenario 2: \((2x-7)^2 > 0\). This occurs when \(x eq 3.5\). In this scenario, we can divide both sides of the inequality by the positive factor \((2x-7)^2\) without changing the direction of the inequality sign:
\[(x-1)(x+2) \leq 0\]

  1. Solving the Quadratic Inequality

Now we focus on the quadratic inequality \((x-1)(x+2) \leq 0\). The roots of the quadratic \(f(x) = (x-1)(x+2)\) are \(x = 1\) and \(x = -2\). These points divide the real number line into three intervals: \((-\infty, -2)\), \([-2, 1]\), and \((1, \infty)\).

  • For \(x < -2\), both \((x-1)\) and \((x+2)\) are negative, so their product is positive.
  • For \(-2 \leq x \leq 1\), the factor \((x+2)\) is non-negative and \((x-1)\) is non-positive, so their product is \(\leq 0\).
  • For \(x > 1\), both factors are positive, so their product is positive.

Thus, the solution to \((x-1)(x+2) \leq 0\) is \(x \in [-2, 1]\).

  1. Conclusion and Final Answer

Combining our results from both scenarios: we have the interval \([-2, 1]\) from the second scenario and the isolated point \(x = 3.5\) from the first scenario. Therefore, the complete set of solutions for the original inequality is the union of these two: \([-2, 1] \cup \{3.5\}\).

Examples

This type of inequality often arises in physics and engineering when determining the range of stability or valid operational values for a system. For instance, if you are designing a support structure where the load capacity depends on an variable \(x\) such that it remains non-positive (representing stable equilibrium), solving this helps identify the specific ranges and critical points for safe operations. Understanding how to handle squared factors ensures you do not miss isolated solutions that also meet the requirement.

Related Questions