Problem 31 - Algebra for College Students

Question

Solve each inequality. $$x^{2}-14 x+49 \geq 0$$

Step-by-Step Solution

Answer

The solution is \(x \geq 7\).

Step 1: Identify the Quadratic

The given inequality is \(x^2 - 14x + 49 \geq 0\). This is a quadratic inequality.

Step 2: Rewrite as Quadratic Equation

Convert the inequality to an equation to find the roots of \(x^2 - 14x + 49 = 0\).

Step 3: Factor the Quadratic

Notice that \(x^2 - 14x + 49\) factors as \((x - 7)^2\). So, \((x - 7)^2 = 0\).

Step 4: Solve for the Root

Set \((x - 7)^2 = 0\), and solve for \(x\). This gives \(x = 7\).

Step 5: Analyze the Quadratic

The quadratic \((x - 7)^2\) is a perfect square, which is always positive or zero. Hence, \((x - 7)^2 \geq 0\) for all \(x\).

Step 6: Specify the Solution

The value \(x = 7\) is the point where \((x - 7)^2\) equals zero. For other values, \((x - 7)^2\) is positive. Therefore, the solution is \(x \geq 7\).

Key Concepts

Quadratic Equations

When we talk about quadratic equations, we mean equations of the second degree, typically in the form of \(ax^2 + bx + c = 0\). Quadratic equations are recognizable because they have an \(x^2\) term. In our example, \(x^2 - 14x + 49\) is a quadratic equation. Understanding how to manage these kinds of equations is fundamental as they frequently appear in algebra.

• The "\(a\)" in the equation represents the coefficient of \(x^2\), which can never be zero.
• The "\(b\)" is the coefficient of \(x\), and "\(c\)" is the constant term.

Quadratic equations can have different types of solutions, which we typically solve by factoring, completing the square, or using the quadratic formula \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\].This process can help determine the roots or solutions of the quadratic equation.

Factoring

Factoring is a key method used to solve quadratic equations. When we "factor" a quadratic, we are expressing it as a product of simpler expressions. This helps in easily finding the roots of the equation. In our problem, the quadratic \(x^2 - 14x + 49\) factors into \((x - 7)^2\).

• Factoring involves looking for two numbers that multiply to \(c\) (in our example, 49) and add to \(b\) (which is -14 here).
• With some quadratics, such as \((x - 7)^2\), the process is straightforward as it immediately shows us that "7" is a special number for this equation.

Once the quadratic is factored, we can set each factor equal to zero to find the solutions.

Perfect Squares

A perfect square in algebra is an expression that can be written in the form \((x - a)^2\). Recognizing perfect squares in quadratic equations simplifies solving a lot since it immediately tells us about the roots.

• In our equation \((x - 7)^2\), "7" is the number that makes the inside of the square zero.
• This tells us that the only root is \(x = 7\), but because it is squared, it represents a repeated root.

Perfect squares are particularly useful in inequalities because they are always non-negative, which means they hold even when equal to zero. In the context of inequalities, this means \((x - 7)^2 \geq 0\) is true for all \(x\).

Solution Sets

A solution set is the collection of all values that satisfy an equation or inequality. In the context of our inequality \((x - 7)^2 \geq 0\), we identified that the only point where \((x - 7)^2\) equals zero is at \(x = 7\).

• For any value greater than 7, \((x - 7)^2\) becomes positive, confirming that \(x \geq 7\) satisfies the inequality.
• Solution sets for quadratic inequalities can often be segments of the x-axis, which include all the values satisfying the condition.

Thus, the solution set for our inequality \((x - 7)^2 \geq 0\) is all \(x\) such that \(x \geq 7\), which is represented in interval notation as \([7, \infty)\). This shows we understand the problem by identifying the interval where the inequality holds true.