91Ó°ÊÓ

Inequalities are mathematical expressions that involve the symbols <, >, ≤, or ≥. They show the relationship between two values where one value is not strictly equal to the other. When solving inequalities, our goal is to find all values of the variable that make the inequality true.

For example, in the inequality \(x > 3\), any value greater than 3 would satisfy the condition. Inequalities can be simple, involving just one variable, or more complex, involving quadratic terms like in our original problem.

To solve inequalities, you generally use similar techniques to those used for solving equations, but there are some special rules:

• When you multiply or divide both sides of an inequality by a negative number, you must reverse the inequality sign.
• You can add or subtract the same value from both sides of the inequality without changing the inequality's direction.

Understanding these basics will help you solve more complex inequalities, like those involving quadratic equations.

A solution set is simply the set of all values that satisfy a given inequality or equation. The solution set can be all real numbers, a specific set of numbers, or even an empty set if no solutions exist.

For example, in our exercise, we determined the solution sets of two inequalities:

• For the inequality \((x+10)^{2} \geq -4\), the solution set is all real numbers, \( (-\infty, \infty)\), because any squared real number is always non-negative and thus greater than or equal to -4.
• For the inequality \((x+10)^{2} \leq -4\), the solution set is the empty set, \(\emptyset\), because no real number squared can be less than zero, hence it cannot be less than or equal to -4.

Using solution sets helps us clearly express and communicate the range of values that satisfy our inequalities.

Quadratic equations are polynomial equations of the form \(ax^{2} + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants and \(a\) is not zero. The solutions to these equations, known as the roots, can be found using various methods like factoring, completing the square, or using the quadratic formula \(x = \frac{{-b \pm \sqrt{{b^{2} - 4ac}}}}{{2a}}\).

In the context of inequalities involving quadratic equations, we are often interested in the values of \(x\) that make the quadratic expression less than, greater than, less than or equal to, or greater than or equal to a certain value.

In our exercise, the quadratic expression \((x+10)^{2}\) plays a key role:

• If we look at \((x+10)^{2} \geq -4\), we know that any square number is at least zero, so this will always be true for all real \(x\), making the solution set \((-\infty, \infty)\).
• For \((x+10)^{2} \leq -4\), it's impossible for any square number (which is always non-negative) to be less than or equal to -4. Therefore, the solution set in this case is \(\emptyset\).

Understanding these core concepts helps us solve and interpret quadratic inequalities accurately.