Problem 32 - Intermediate Algebra

Question

Solve each inequality. $$ -4\left(x^{2}-36\right) \geq 0 $$

Step-by-Step Solution

Answer

The solution is \([-6, 6]\).

Step 1: Simplify the Inequality

The inequality given is \[-4(x^2 - 36) \geq 0.\] First, distribute the \(-4\) inside the parentheses:\[-4x^2 + 144 \geq 0.\] This is the simplified form of the inequality.

Step 2: Rearrange to Standard Form

Rearrange the inequality to bring terms on one side by adding \(4x^2\) to both sides:\[144 \geq 4x^2.\]Now divide all terms by 4 to further simplify:\[36 \geq x^2.\] This can also be written as \(x^2 \leq 36\).

Step 3: Solve for x

To solve \(x^2 \leq 36\), take the square root of both sides. Recall square roots yield both positive and negative solutions:\(-\sqrt{36} \leq x \leq \sqrt{36}.\)This gives:\(-6 \leq x \leq 6.\)

Step 4: Express the Solution

The solution to the inequality is the set of all \(x\) values such that:\[-6 \leq x \leq 6.\]This is the interval notation \([-6, 6]\).

Key Concepts

Quadratic Inequalities

Quadratic inequalities are like equations but instead of an equal sign, they use inequality symbols like \(\geq, \leq, >, \text{or} <\). These inequalities often involve quadratic expressions, which are polynomials of the form \(ax^2 + bx + c\). A typical problem involves determining the range of values for \(x\) that satisfy the inequality.

To solve quadratic inequalities, you generally follow these steps:

• Simplify the inequality, if necessary.
• Set the quadratic expression equal to zero to find critical points, which are the values where the expression changes sign.
• Determine the intervals on the number line that satisfy the inequality by testing values from each interval.
• Use the correct inequality sign to indicate the solution set.

In our exercise, we simplified the inequality first, then expressed it in a form where we could solve for \(x\) directly, which made the process straightforward.

Interval Notation

Interval notation is a way of representing the solution set of inequalities, which shows the range of possible values for a variable. It's compact and precise, making it a preferred method to display solutions.

Intervals can be

• closed - including the endpoints (use brackets [ ]),
• open - excluding the endpoints (use parentheses ( )),
• or a mix of both.

For instance, in the exercise, the solution to the inequality \(-6 \leq x \leq 6\) is expressed as the interval \([-6, 6]\). This tells us that \(x\) can be any real number between \(-6\) and \(6\), inclusive.

Solving Inequalities

Solving inequalities involves finding all possible values of a variable that make the inequality true. The process can be distinct from solving equations because inequality signs often require specific attention and manipulation.

Here's an approach to solving inequalities:

• First, simplify or rearrange the inequality as needed. Ensure all terms are on one side.
• If dealing with quadratic expressions, factor or solve the related equality to identify critical points.
• Choose test points between the critical values to determine which sections satisfy the inequality.
• Finally, express the solution in interval notation if required.

In our step-by-step solution, we simplified the inequality and solved for \(x\) by using algebraic manipulation like distributing, rearranging, and extracting square roots. This resulted in a solution that is easy to understand and express as an interval, ensuring all values in that range satisfy the original inequality.