Radical inequalities are inequalities in which the unknown appears under a radical sign. This is a fundamental topic in algebra, as it requires the simultaneous use of properties of radicals, sign analysis, and domain conditions.
Unlike polynomial or rational inequalities, radical inequalities cannot be handled through algebraic manipulation alone to manipulate the expression algebraically: every step must respect the domain of the radicals involved.
In particular, when squaring both sides, one must carefully verify that this transformation is logically equivalent to the original inequality. Improper use of squaring can introduce extraneous solutions.
We will cover:
- the domain conditions for radicals;
- the general solution method;
- the fundamental cases;
- inequalities with a single radical;
- inequalities with multiple radicals;
- the most common mistakes to avoid.
Table of Contents
- What Are Radical Inequalities
- Domain Conditions
- Inequalities of the Type \(\sqrt{A(x)}>B(x)\)
- Inequalities of the Type \(\sqrt{A(x)}<B(x)\)
- Inequalities with Radicals on Both Sides
- Inequalities with Multiple Radicals
- General Method
- Common Mistakes to Avoid
What Are Radical Inequalities
A radical inequality is an inequality in which the unknown appears under a radical sign.
For example:
\[ \sqrt{x-1}>2 \]
\[ \sqrt{2x+3}\le x \]
\[ \sqrt{x+1}>\sqrt{2x-3} \]
are all radical inequalities.
The most delicate aspect of these inequalities is that real square roots exist only when the radicand is greater than or equal to zero.
For this reason, before any algebraic manipulation, one must always determine the domain conditions.
Domain Conditions
Whenever a square root appears:
\[ \sqrt{A(x)}, \]
the following must necessarily hold:
\[ A(x)\ge 0. \]
This is the fundamental domain condition.
Example
Consider:
\[ \sqrt{2x-5}>1. \]
The radical is defined only if:
\[ 2x-5\ge 0. \]
Solving:
\[ 2x\ge 5 \]
we obtain:
\[ x\ge \frac52. \]
This means that any final solution must belong to the interval:
\[ \left[\frac52,+\infty\right). \]
Inequalities of the Type \(\sqrt{A(x)}>B(x)\)
Consider an inequality of the form:
\[ \sqrt{A(x)}>B(x). \]
The approach depends on the sign of the right-hand side.
Since the square root is always non-negative:
\[ \sqrt{A(x)}\ge 0, \]
it follows that:
- if \(B(x)<0\), the inequality is automatically satisfied whenever the radical is defined;
- if \(B(x)\ge 0\), both sides may be squared.
Example
Solve:
\[ \sqrt{x+1}>3. \]
First, we impose the domain conditions:
\[ x+1\ge 0, \]
giving:
\[ x\ge -1. \]
The right-hand side is positive. We may therefore square both sides:
\[ x+1>9. \]
Hence:
\[ x>8. \]
Intersecting with the domain conditions, we obtain:
\[ S=(8,+\infty). \]
Inequalities of the Type \(\sqrt{A(x)}<B(x)\)
Now consider:
\[ \sqrt{A(x)}<B(x). \]
This case requires even more care.
Since the radical is always non-negative, for a non-negative quantity to be less than \(B(x)\), we need:
\[ B(x)>0. \]
Only after imposing this condition may we square both sides.
The equivalent system is therefore:
\[ \begin{cases} A(x)\ge 0 \\ B(x)>0 \\ A(x)<B(x)^2 \end{cases} \]
Example
Solve:
\[ \sqrt{x-2}<x-4. \]
We impose the conditions:
\[ \begin{cases} x-2\ge 0 \\ x-4>0 \end{cases} \]
that is:
\[ \begin{cases} x\ge 2 \\ x>4 \end{cases} \]
The second condition already implies the first, so it suffices to require:
\[ x>4. \]
We may now square both sides:
\[ x-2<(x-4)^2. \]
Expanding:
\[ x-2<x^2-8x+16. \]
Rearranging:
\[ 0<x^2-9x+18, \]
that is:
\[ x^2-9x+18>0. \]
Factoring:
\[ x^2-9x+18=(x-3)(x-6). \]
The inequality becomes:
\[ (x-3)(x-6)>0. \]
From the sign analysis we obtain:
\[ x<3 \quad \text{or} \quad x>6. \]
Intersecting with the condition \(x>4\), we are left with:
\[ S=(6,+\infty). \]
Inequalities with Radicals on Both Sides
Consider inequalities of the form:
\[ \sqrt{A(x)}>\sqrt{B(x)}. \]
In this case, both radicals must be defined:
\[ \begin{cases} A(x)\ge 0 \\ B(x)\ge 0 \end{cases} \]
Once these conditions are imposed, we may square both sides:
\[ A(x)>B(x). \]
Example
Solve:
\[ \sqrt{x+3}>\sqrt{2x-1}. \]
We impose the domain conditions:
\[ \begin{cases} x+3\ge 0 \\ 2x-1\ge 0 \end{cases} \]
that is:
\[ \begin{cases} x\ge -3 \\ x\ge \frac12 \end{cases} \]
Therefore:
\[ x\ge \frac12. \]
We now square both sides:
\[ x+3>2x-1. \]
Solving:
\[ 4>x, \]
that is:
\[ x<4. \]
Intersecting with \(x\ge \frac12\), we obtain:
\[ S=\left[\frac12,4\right). \]
Inequalities with Multiple Radicals
When an inequality contains multiple radicals in the same expression, the procedure may require repeated squaring.
In such cases, it is essential to:
- isolate one radical at a time;
- always impose the domain conditions;
- check the final solutions.
Example
Solve:
\[ \sqrt{x+5}-\sqrt{x-1}>0. \]
We impose the domain conditions:
\[ \begin{cases} x+5\ge 0 \\ x-1\ge 0 \end{cases} \]
giving:
\[ \begin{cases} x\ge -5 \\ x\ge 1 \end{cases} \]
Therefore:
\[ x\ge 1. \]
We move one radical to the right-hand side:
\[ \sqrt{x+5}>\sqrt{x-1}. \]
Both sides are non-negative, so we may square both sides:
\[ x+5>x-1. \]
Simplifying:
\[ 5>-1. \]
This is always true.
Therefore, the inequality holds for all values in the domain:
\[ S=[1,+\infty). \]
General Method
To solve a radical inequality correctly, it is advisable to follow a systematic approach.
- Determine the domain conditions for the radicals.
- Isolate a radical if necessary.
- Analyze the signs of both sides of the inequality.
- Square both sides only when equivalence is guaranteed.
- Solve the resulting inequality.
- Intersect with the initial conditions.
- Check for any extraneous solutions.
The final check is especially important for inequalities obtained after repeated squaring.
Common Mistakes to Avoid
Forgetting the Domain Conditions
This is the most common mistake.
For example:
\[ \sqrt{x-2}>1 \]
necessarily requires:
\[ x-2\ge 0. \]
Ignoring this condition can lead to invalid solutions.
Squaring Without Checking the Sign
The inequalities:
\[ a>b \]
and
\[ a^2>b^2 \]
are not equivalent in general.
Squaring preserves equivalence only under appropriate sign conditions.
Not Checking the Final Solutions
After squaring, extraneous solutions may appear.
For this reason, the final result must always be verified against the original inequality.
Radical inequalities require a rigorous and systematic approach. Every step must respect both the domain of the radicals and the conditions that ensure the equivalence of the transformations applied.
The key point is not simply knowing how to square both sides, but understanding when doing so is logically valid.
A solid command of radical inequalities is indispensable for tackling the study of functions, curve intersections, and more advanced inequalities in mathematical analysis.