Solving the Inequality x^2 5x ≤ -4

Mathematical inequalities can often be solved through a variety of methods, including algebraic manipulations, factoring, and graphing. In this article, we will explore the process of solving the given inequality:

Step-by-Step Solution of the Inequality

To begin, we start with the given inequality:

x^2 5x ≤ -4

Step 1: Rearrange the Inequality

First, we rearrange the inequality to standard form by adding 4 to both sides of the equation:

x^2 5x 4 ≤ -4 4

Simplifying the right side, we get:

x^2 5x 4 ≤ 0

Step 2: Factor the Quadratic Expression

Next, we factor the quadratic expression on the left side of the inequality. We need to find two numbers that multiply to 4 and add up to 5. These numbers are 1 and 4. Thus, we can factor the expression as follows:

(x 1)(x 4) ≤ 0

Step 3: Solve the Factored Inequality

To solve the factored inequality, we determine the critical points by setting each factor equal to zero:

x 1 0 → x -1

x 4 0 → x -4

These values divide the number line into three intervals. We will test each interval to determine where the inequality is satisfied.

Testing Intervals

- For x , choose x -5:

(-5 1)(-5 4) (-4)(-1) 4 > 0

- For -4 ≤ x ≤ -1, choose x -2.5:

(-2.5 1)(-2.5 4) (-1.5)(1.5) -2.25 ≤ 0

- For x > -1, choose x 0:

(0 1)(0 4) (1)(4) 4 > 0

Since we are looking for the intervals where the expression is less than or equal to zero, the solution is the interval where the product is non-positive, which is -4 ≤ x ≤ -1.

Graphical Solution

Graphing the inequality helps visualize the solution. Consider the parabolic function:

y x^2 5x 4

The roots of the quadratic equation are x -1 and x -4. We plot the parabola and find the region where the parabola is below the horizontal line y -4

This plot shows the area below the horizontal line y -4 over the interval x ∈ [-4, -1]. It fills a cup from the vertex upward to y -4.

Final Answer

The solution to the inequality x^2 5x ≤ -4 is:

-4 ≤ x ≤ -1