Problem 54 - Algebra 1

Question

CHECKING SOLUTIONS OF INEQUALTTIES Check whether the given number is a solution of the inequality. $$x^{2}-10>16 ; 6$$

Step-by-Step Solution

Answer

Yes, the given number 6 is a solution of the inequality \(x^{2}-10>16\)

Step 1: Simplification

The inequality to be simplified is \(x^{2}-10>16\). Subtract 16 from both sides to get it into the general format of a quadratic inequality. The expression simplifies to \(x^{2}-26>0\).

Step 2: Substitute the given value

Now, replace the x in the inequality \(x^{2}-26>0\) with the given number 6. The substitution gives \((6)^{2}-26>0\). Simplifying the left hand side results in \(36-26>0\), yielding \(10>0\).

Step 3: Verification

The final step is to verify the inequality 10>0. The statement is true since 10 is indeed greater than 0.

Key Concepts

Quadratic Inequalities

Quadratic inequalities, such as the one expressed by the equation \(x^{2}-10>16\), are algebraic expressions where a quadratic polynomial is set against a constant or another quadratic polynomial, using inequality signs like >, <, \(\geq\), or \(\leq\). These inequalities hold significant importance as they pop up frequently in mathematical analysis and calculus, as well as in various real-world problems such as physics and economics.

To solve these inequalities, one typically brings all terms to one side of the inequality to create a standard form (like \(x^{2} - 26 > 0\)) and then identifies the roots of the corresponding quadratic equation \(x^{2} - 26 = 0\). The solutions to the inequality are ranges of x-values that make the inequality true, and the roots help to determine these ranges. Factoring, completing the square, or using the quadratic formula are common techniques used to find these roots. Once the critical points are found, test intervals between the roots to see whether the original inequality holds true in those ranges.

Inequality Substitution

The process of inequality substitution involves replacing the variable in an inequality with a specific value to determine if the inequality holds true. In the given exercise, we used the number 6 as the substitute for x in the quadratic inequality \(x^{2}-26>0\). This step is critical as it directly tests the validity of the potential solution.

To substitute effectively, follow these simple steps: Replace every instance of the variable with the given number, as we did with the substitution that resulted in \( (6)^{2}-26>0 \). Simplify the expression and compare the result to determine if the inequality sign is indeed respected.

Why is substitution necessary?

Substitution acts as the acid test to check if our suspected solution really satisfies the inequality. Without this step, we can't be sure whether the proposed number is an actual solution to the problem.

Inequality Simplification

Inequality simplification is a fundamental step in solving inequalities, aimed at making the original problem more manageable. The process often involves combining like terms and moving all the terms to one side of the inequality to isolate the quadratic term.

In our example of \(x^{2}-10>16\), the first step to find a simplified form is to subtract 16 from both sides to obtain \(x^{2}-26>0\). This makes the inequality easier to analyze because it transforms into a recognizable format, where one side is set to zero.

Benefits of Simplification

Simplifying the inequality removes extraneous information, which could distract from the main issue—finding the range of values for the variable that make the inequality true.

• This step lowers the risk of computational errors in further operations.
• It prepares the expression for substitution or other methods like factoring or graphing.

Remember, the end goal is to have a clear, simplified inequality that can then be solved or verified using appropriate methods.