Question

Solve the inequality algebraically. \(x^2-11 x \geq-28\)

Short answer

The solution to the inequality \(x^2-11 x \geq -28\) is \(x \leq 4\) and \(x \geq 7\).

Step-by-step solution

Convert Inequality to Equation

Convert the inequality \(x^2-11 x \geq-28\) into quadratic equation, i.e. \(x^2-11x+28=0\)

Find the Roots

Factor the quadratic equation to find its roots. The equation can be factored into \((x-7)(x-4)=0\). Thus the roots are \(x=7\) and \(x=4\)

Test the intervals

Now, divide the number line into intervals based on these roots and test each interval in the inequality. The intervals are \((-∞, 4)\), \((4, 7)\), and \((7, +∞)\). Pick a test point from each interval and substitute it into the inequality. If the inequality holds true for that test point, then the entire interval is part of the solution. If it does not hold true, the entire interval is not part of the solution.

Find the Solution for the Inequality

After testing the intervals, we found that the solutions of the inequality \(x^2-11 x \geq-28\) are \(x \leq 4\) and \(x \geq 7\).

Key concepts

Factoring Quadratic Equations

Factoring quadratic equations is a crucial skill in solving many types of algebraic problems, including quadratic inequalities. When you factor a quadratic equation like \(ax^2 + bx + c = 0\), you're breaking it down into simpler expressions that can be multiplied to get the original equation. This process allows us to easily find the values of \(x\) that make the equation true, known as the "roots" of the equation. For example, if you have \(x^2 - 11x + 28 = 0\), you would look for two numbers that multiply to 28 and add to -11. These numbers are -4 and -7, thus the equation factors into \((x-7)(x-4)=0\). This format makes it easy to identify the roots at \(x = 7\) and \(x = 4\). Factoring helps not only to solve quadratic equations directly but also to prepare them for further analysis like interval testing.

Roots of Equations

The roots of an equation are the values of \(x\) that satisfy the equation \(ax^2 + bx + c = 0\). Once you have factored the quadratic equation, finding the roots becomes straightforward. Each factor set equal to zero gives a root. For instance, with \((x-7)(x-4)=0\), you can solve each parentheses separately to find that \(x-7=0\) results in \(x=7\) and \(x-4=0\) results in \(x=4\). These roots are the points where the graph of the quadratic equation intersects the x-axis. Knowing the roots is essential for solving quadratic inequalities because they divide the number line into sections, which are then analyzed separately.

Interval Testing

Interval testing is a method used to determine which parts of the number line satisfy a given inequality. After identifying the roots of the quadratic equation, we use them to break the number line into intervals. For example, with roots at 7 and 4, the intervals are

• \((-∞, 4)\)
• \((4, 7)\)
• \((7, ∞)\)

Next, pick a test point from each interval and plug it back into the original inequality \(x^2 - 11x \geq -28\). This helps check if the inequality holds true within that interval. If it does, the entire interval satisfies the inequality. For example, choosing 0 from \((-∞, 4)\), 5 from \((4, 7)\), and 8 from \((7, ∞)\), you will find which intervals satisfy the inequality. This method is essential for painting a complete picture of the solution set.

Inequality Solutions

Finding the solution to a quadratic inequality involves synthesizing all previous steps. You have factored the quadratic to find its roots, broken the number line into intervals, and used interval testing to check each territory. The solution to the inequality \(x^2 - 11x \geq -28\) results in identifying the values of \(x\) that make the inequality true. In this case, testing displays that intervals \(x \leq 4\) and \(x \geq 7\) satisfy the inequality. This means that for any \(x\) within these intervals, the inequality holds. Delivering solutions in inequality form ensures clarity and comprehensiveness in understanding which values make the inequality true.