Problem 31 - Precalculus : Building Concepts and Connections

Question

Solve the inequality algebraically or graphically. $$x^{2}-4 \geq x$$

Step-by-Step Solution

Answer

The solution for the inequality \(x^{2} - 4 \geq x\) is \(- \infty < x \leq -2\) and \(2 \leq x < + \infty\)

Step 1: Rearrange the inequality

Rearrange the inequality by moving all the terms on one side to form a quadratic equation. So, the inequality becomes \(x^{2} - x - 4 \geq 0\).

Step 2: Factor the quadratic equation

Factor the quadratic equation to its simplest form which helps to find the roots of the equation. The factored form of \(x^{2} - x - 4\) is \((x - 2)(x + 2) \geq 0\).

Step 3: Find the roots

Setting each factor equal to zero grants the roots of the equation. In this case, it gives \(x - 2 = 0\) and \(x + 2 = 0\), thus obtaining the roots, \(x = 2\) and \(x = -2\).

Step 4: Determine the solution

Since this is a quadratic equation of inequality, determine the solution by performing a number line test at values including and between the roots. Insert any number below -2, any number between -2 and 2, and any number above 2 into \(x^{2} - x - 4\). Whichever sections give values greater than or equal to 0 will be part of the solution set. Testing these ranges yield the solution as \(- \infty < x \leq -2\) and \(2 \leq x < + \infty\).

Key Concepts

Factoring Quadratic Equations

Factoring a quadratic equation is a critical skill in algebra. Essentially, it involves rewriting the quadratic in its simplest form by finding its roots or solutions. The equation in our example is \(x^2 - x - 4 = 0\). To factor it, you need to express it as a product of two binomials: \((x - 2)(x + 2)\).

Factoring is mostly about finding two numbers that add up to the coefficient of the linear term (in this case, -1) and multiply to the constant term (-4). It helps you identify points where the function equals zero, known as roots, which are essential in solving inequalities.

• Set each factor equal to zero to find the roots: \(x - 2 = 0\) and \(x + 2 = 0\).
• The roots will be \(x = 2\) and \(x = -2\).

Factoring is valuable for simplifying the solution process.

Solving Inequalities

Solving inequalities involves determining the values of \(x\) that make the inequality true. Inequalities express a relationship where one side is not necessarily equal to the other, symbolized by \(>\), \(<\), \(\geq\), or \(\leq\).

For \(x^2 - x - 4 \geq 0\), you're looking for where the expression is greater than or equal to zero. Once factored, you have \((x - 2)(x + 2) \geq 0\).

• Identify intervals created by the roots: between and around \(-2\) and \(2\).
• Determine which intervals satisfy the inequality by checking test points.

Solving quadratic inequalities gives you the range of possible solutions rather than a single answer.

Number Line Test

The number line test is an intuitive way to determine which intervals satisfy a quadratic inequality. After factoring the equation, plot the roots on a number line. This creates intervals to test where the inequality holds true.

In this exercise, the roots are \(-2\) and \(2\), creating three intervals: \((-\infty, -2)\), \((-2, 2)\), and \((2, +\infty)\).

• Choose a test point from each interval, such as \(-3\), \(0\), and \(3\).
• Substitute these points back into the inequality.
• See if the resulting expression is greater than or equal to zero.

This test helps confirm which intervals are part of the solution set.

Algebraic Manipulation

Algebraic manipulation is the process of rearranging or simplifying equations and inequalities to uncover solutions. In our example, start by moving all terms to one side to get \(x^2 - x - 4 \geq 0\).

This method makes it easier to factor the equation and proceed with solving.

• Rewriting terms is essential to set up the equation for factoring.
• Combine like terms and use operations systematically to simplify.

Manipulating equations is a valuable skill for solving more complex problems efficiently. By practicing these steps, you gain greater control over identifying and working through algebraic expressions.