Questions: UNIT 4 - CHALLENGE 4.2: The solution set of the quadratic inequality x^2 - 5x + 4 ≤ 0 is a.) -4 ≤ x ≤ -1 b.) x ≤ -4 or x ≥ -1 c.) 1 ≤ x ≤ 4 d.) x ≤ 1 or x ≥ 4

Transcript text: UNIT 4 - CHALLENGE 4.2: The solution set of the quadratic inequality $x^{2}-5 x+4 \leq 0$ is $\qquad$ a.) $-4 \leq x \leq-1$ b.) $x \leq-4$ or $x \geq-1$ c.) $1 \leq x \leq 4$ d.) $x \leq 1$ or $x \geq 4$

Solution

Solution Steps

To solve the quadratic inequality $x^2 - 5x + 4 \leq 0$, we first find the roots of the corresponding quadratic equation $x^2 - 5x + 4 = 0$. These roots will help us determine the intervals to test for the inequality. We then test the intervals between and outside the roots to see where the inequality holds true.

Step 1: Find the Roots of the Quadratic Equation

To solve the inequality $x^2 - 5x + 4 \leq 0$, we first find the roots of the equation $x^2 - 5x + 4 = 0$. Solving this equation, we find the roots to be $x = 1$ and $x = 4$.

Step 2: Determine the Intervals

The roots divide the number line into intervals. We have the intervals $(-\infty, 1)$, $(1, 4)$, and $(4, \infty)$.

Step 3: Test the Intervals

We test the sign of the quadratic expression $x^2 - 5x + 4$ within each interval:

For the interval $(1, 4)$, choose a test point, such as $x = 2$. Substituting into the expression gives $2^2 - 5 \times 2 + 4 = 4 - 10 + 4 = -2$, which is less than or equal to zero.
For the intervals $(-\infty, 1)$ and $(4, \infty)$, the expression is positive.

Step 4: Include the Endpoints

Since the inequality is $\leq$, we include the endpoints where the expression equals zero. At $x = 1$ and $x = 4$, the expression evaluates to zero.