Question

If \(a>0,\) show that the solution set of the inequality $$ x^{2}>a $$ consists of all numbers \(x\) for which $$ x<-\sqrt{a} \text { or } x>\sqrt{a} $$

Short answer

The solutions are \[ x < -\sqrt{a} \] or \[ x > \sqrt{a} \].

Step-by-step solution

Understand the inequality

The given inequality is \[ x^2 > a \]. This inequality states that the square of any number x must be greater than the positive number a.

Rewrite the inequality in another form

To solve for x, take the square root of both sides of the inequality. This provides two cases: \[ x^2 > a \]. When you take the square root of both sides, you need to consider both the positive and negative square roots. The inequality becomes:\[ |x| > \sqrt{a} \].

Split the absolute value into two inequalities

The inequality \[ |x| > \sqrt{a} \] can be rewritten without the absolute value as two separate inequalities:\[ x > \sqrt{a} \] and \[ x< -\sqrt{a} \].

Combine the solutions

The solution set for the inequality \[ x^2 > a \] is the combination of the two inequalities from the previous step. Therefore, the solution is:\[ x < -\sqrt{a} \] or \[ x > \sqrt{a} \].

Key concepts

Quadratic Inequalities

Quadratic inequalities involve expressions where a quadratic polynomial is compared to a value, usually zero, using inequality symbols like <, >, ≤, or ≥. In this case, we have \(x^2 > a\). To solve such inequalities, first rewrite the inequality in a form that is easier to understand. For a positive constant \(a\), the quadratic inequality can be transformed into an absolute value inequality, which simplifies the problem significantly.
Here's the breakdown:

Rewrite \(x^2 > a\) as \(|x| > \sqrt{a}\)\.

This tells us that the distance between \(x\) and zero needs to be greater than \(\text{\textbackslash sqrt{a}}\).

This cleverly uses the property that squaring both sides of an inequality involving positive numbers does not change the inequality direction.

Absolute Value Inequalities

Absolute value inequalities express a condition on the distance of a variable from zero. The notation \(|x| > b\) means that the value of \(x\) is more than \(b\) units away from zero, either in the positive or negative direction.
Here's how you handle \(|x| > \sqrt{a}\):

Break it into two separate inequalities: \(x < -\text{\textbackslash sqrt{a}}\) and \(x > \sqrt{a}\).

This step separates the inequality into the parts that the absolute value expression represents.

Remember that the absolute value essentially asks how far away a number is from zero, but we need the exact regions that satisfy this condition.

Solution Sets

When solving inequalities, the goal is to determine the set of all possible values that satisfy the inequality.
In our case, for the inequality \(x^2 > a\):

After simplifying and considering both parts of the inequality \(x < -\text{\textbackslash sqrt{a}}\) and \(x > \sqrt{a}\), we combine them to get the solution set.

The combined solution represents all values of \(x\) that make the original inequality true. It specifically means that \(x\) is in the ranges where either \(x\) is less than the negative square root of \(a\), or more than the positive square root of \(a\).

This can be visualized on a number line as two separate regions: one extending to the left of \(-\text{\textbackslash sqrt{a}}\), and one extending to the right of \( \ \text{\textbackslash sqrt{a} \).