Problem 43 - Algebra for College Students

Question

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

Step-by-Step Solution

Answer

\(x \leq 0\) or \(x \geq 2\)

Step 1: Factor the Quadratic Expression

Begin by factoring the quadratic expression. The expression given is \(x^2 - 2x\). Factoring out the common term, we have: \[ x(x - 2) \geq 0 \]

Step 2: Identify Critical Points

The solutions to the equation \(x(x-2) = 0\) give us the critical points. These occur where the expression equals zero. so \(x = 0\) or \(x = 2\) are the critical points.

Step 3: Test Intervals Between and Beyond Critical Points

The number line is divided into intervals by the critical points: \((-\infty, 0)\), \((0, 2)\), and \((2, \infty)\). Test a value from each interval in the inequality \(x(x - 2) \geq 0\).- For \((-\infty, 0)\), test \(x = -1\): \((-1)(-1 - 2) = 3 \geq 0\), false- For \((0, 2)\), test \(x = 1\): \(1(1 - 2) = -1 \geq 0\), false- For \((2, \infty)\), test \(x = 3\): \(3(3 - 2) = 3 \geq 0\), true

Step 4: Conclusion from Critical Points and Test Intervals

Include the endpoints since the inequality is non-strict (\(\geq\)). The solution includes the intervals where the test returns true, including the critical points themselves (where the expression equals 0): The solution is \(x \leq 0\) or \(x \geq 2\).

Key Concepts

Quadratic Inequalities

Quadratic inequalities involve expressions of degree two, and they state a relationship comparing a quadratic expression to zero. For example, an inequality like \(x^2 - 2x \geq 0\) requires us to find the set of \(x\) values that satisfy the inequality. Solving these involves several steps:

• Factoring: Begin by writing the quadratic expression in a factored form. This makes it easier to analyze and determine where the expression changes sign.
• Inequality Sign: Pay attention to the inequality sign. Here, the \(\geq\) sign indicates a non-strict inequality, meaning we are interested in when the expression is zero or positive.

The solution to a quadratic inequality can be found graphically as well but solving algebraically allows you to determine exact interval solutions. When you solve the inequality, you're finding out where the quadratic expression is zero (equal to), positive (greater than), or in some cases, negative (less than) depending on the inequality presented.

Critical Points

Critical points are locations on the number line where the expression equals zero, which divides the number line into different segments. These are found by setting the quadratic expression in its factored form equal to zero. In the inequality \(x^2 - 2x \geq 0\) after factoring, we get \(x(x-2) = 0\). Thus, the critical points are:

• \(x = 0\): This is one point where the expression switches sign or is equal to zero.
• \(x = 2\): This is another point where the expression equals zero.

These points are important because any changes in the inequality, such as from positive to negative, happen around these points. Recognizing them helps set up the intervals over which you will test the inequality to find your solution. Critical points often also become part of the solution, especially when dealing with non-strict inequalities.

Test Intervals

After identifying critical points, the number line is divided into intervals. In our example involving \(x(x-2) \geq 0\), three significant intervals emerge: \((-\infty, 0)\), \((0, 2)\), and \((2, \infty)\). Solving the inequality requires testing points from each interval to see if they satisfy the inequality:

• Test Point in \((-\infty, 0)\): Choose \(x = -1\). After substituting, \((-1)(-3) = 3\), which is not \(\geq 0\), thus false for this interval.
• Test Point in \((0, 2)\): Choose \(x = 1\). Substituting gives \(1(1-2) = -1\), which is also not \(\geq 0\), thus also false.
• Test Point in \((2, \infty)\): Choose \(x = 3\). Substituting gives \(3(3-2) = 3\), which satisfies \(\geq 0\), hence true.

By testing points within these intervals, you can determine in which intervals the inequality holds true. Remember that the critical points themselves are included in the solution if the original inequality is non-strict (\(\geq\) or \(\leq\)). Therefore, for \(x^2 - 2x \geq 0\), the solution is \(x \leq 0\) or \(x \geq 2\). This method ensures that no interval where the inequality holds is overlooked.