Determine the values for which $x^2 - 7x + 10 \leq 0$

Analysis

To solve the inequality $x^2 - 7x + 10 \leq 0$, we need to find the range of values for $x$ that makes the expression less than or equal to zero. This is a quadratic inequality.

Concept

1. Factorize the quadratic expression: We look for two numbers that multiply to give $+10$ and add up to $-7$. These numbers are $-2$ and $-5$.
2. Find the critical points: Set the factors to zero to find where the expression equals zero.
3. Test the intervals: Determine which region between the critical points satisfies the "less than or equal to" condition.

Step-by-Step Solution

Step 1: Factorize the quadratic $x^2 - 7x + 10 = 0$ $(x - 2)(x - 5) = 0$

Step 2: Identify the critical points $x - 2 = 0 \implies x = 2$ $x - 5 = 0 \implies x = 5$

Step 3: Determine the interval We have three regions on the number line: $x < 2$, $2 \leq x \leq 5$, and $x > 5$.

• If we pick a number between 2 and 5 (e.g., $x = 3$): $(3 - 2)(3 - 5) = (1)(-2) = -2$. Since $-2 \leq 0$, this region satisfies the inequality.
• If we pick a number less than 2 (e.g., $x = 0$): $(0 - 2)(0 - 5) = (-2)(-5) = +10$. Since $10$ is not $\leq 0$, this region does not satisfy the inequality.
• If we pick a number greater than 5 (e.g., $x = 6$): $(6 - 2)(6 - 5) = (4)(1) = +4$. Since $4$ is not $\leq 0$, this region does not satisfy the inequality.

Therefore, the inequality holds true for $2 \leq x \leq 5$.

Conclusion

The values for which the expression is less than or equal to zero are between 2 and 5 inclusive.

Exam Tip: For quadratic inequalities of the form $(x - a)(x - b) \leq 0$ (where $a < b$), the solution is always the interval between the roots: $a \leq x \leq b$. If the inequality were $\geq 0$, the solution would be $x \leq a$ or $x \geq b$.