Problem 49 - Algebra 2

Question

Solve each inequality. $$ x^{2}-4 x \leq 0 $$

Step-by-Step Solution

Answer

The solution to the inequality is \([0, 4]\).

Step 1: Factor the Quadratic Expression

The inequality given is \( x^2 - 4x \leq 0 \). First, we factor the expression \( x^2 - 4x \). Factor \( x \) from both terms to get \( x(x - 4) \). This gives us the inequality: \( x(x - 4) \leq 0 \).

Step 2: Find Critical Points

To solve the inequality \( x(x - 4) \leq 0 \), we first find the critical points where the expression is equal to zero. Set \( x(x - 4) = 0 \). Solving this equation gives the critical points: \( x = 0 \) and \( x = 4 \).

Step 3: Test Intervals Between Critical Points

The critical points divide the number line into intervals: \((-fty, 0)\), \((0, 4)\), and \((4, fty)\). We test a value from each interval to determine where the product \( x(x - 4) \) is less than or equal to zero. **For \((-fty, 0)\), test \( x = -1 \):** \(-1(-1 - 4) = 5\) (positive). **For \((0, 4)\), test \( x = 2 \)**: \(2(2 - 4) = -4\) (negative). **For \((4, fty)\), test \( x = 5 \):** \(5(5 - 4) = 5\) (positive).

Step 4: Determine Solution Including Boundary Points

The inequality \( x(x - 4) \leq 0 \) holds true where the product is negative or zero. From the test intervals, the inequality is satisfied in \( [0, 4] \). Include the boundary points \( x = 0 \) and \( x = 4 \) because the inequality is non-strict (\( \leq \)). For both points, \( x(x - 4) = 0 \) is satisfied.

Key Concepts

Factoring Quadratics

The first step to solving the inequality \(x^2 - 4x \leq 0\) is to factor it. Factoring quadratics involves expressing the equation as a product of its simpler linear factors. In our case, notice that both terms share a common factor \(x\). By factoring out \(x\), we simplify the expression to \(x(x - 4) \). This means that we have rewritten the quadratic as a product of its factors: \(x\) and \(x-4\).
Factoring is a useful technique because it transforms the equation into a multiplication of simpler expressions. This makes it easier to see where the product equals zero, which is crucial for solving inequalities.

Critical Points

Once the quadratic expression is factored, the next step is to find the critical points. Critical points are the values of \(x\) that make the product of factors equal zero. To find them, solve the equation \(x(x - 4) = 0\).
In this scenario, the product can be zero if either \(x = 0\) or \(x - 4 = 0\). Solving \(x - 4 = 0\) gives us \(x = 4\). Thus, the critical points are at \(x = 0\) and \(x = 4\).
These points are crucial because they divide the number line into distinct intervals that we will test to find where the inequality holds true.

Test Intervals

With the critical points identified, the number line is split into intervals:

• \((-\infty, 0)\)
• \((0, 4)\)
• \((4, \infty)\)

We need to test values within these intervals to see if they satisfy the inequality \(x(x - 4) \leq 0\). For each interval, choose a test point:

• In \((-\infty, 0)\), use \(x = -1\). This gives \(-1(-1 - 4)\), which equals \(5\), a positive number.
• In \((0, 4)\), use \(x = 2\). This gives \(2(2 - 4)\), which equals \(-4\), a negative number.
• In \((4, \infty)\), use \(x = 5\). This gives \(5(5 - 4)\), which equals \(5\), a positive number.

This testing helps determine where the product of the factors is less than or equal to zero, guiding us to the solution of the inequality.

Inequality Solution

After testing the intervals, we can determine where the inequality \(x(x - 4) \leq 0\) is satisfied. The inequality holds in the interval where the product of the factors is negative or zero, proving that \([0, 4]\) is the solution.
Importantly, because the inequality is non-strict (\(\leq\)), we also include the critical points \(x = 0\) and \(x = 4\) in the solution. At these points, the product \(x(x - 4) = 0\), which fulfills the \(\leq 0\) condition.
So, the complete solution to the inequality is the closed interval \([0, 4]\), where every point within this range satisfies the inequality.