Irrational equations and Inequalities: Problems with Solutions

Irrational equations and Inequalities: Problems with Solutions

Edited By Komal Miglani | Updated on Jul 02, 2025 08:08 PM IST

Inequalities are mathematical expressions showing the relationship between two values, indicating that one value is greater than, less than, or not equal to another. Understanding inequalities is crucial for solving various mathematical problems, from basic arithmetic to advanced calculus. Irrational equations are help in solving fractional power equations.

This Story also Contains
  1. Inequalities
  2. Types of Inequalities
  3. Irrational equation
  4. Irrational Inequalities:
  5. Solved Examples Based On Irrational Inequalities and Equations:
Irrational equations and Inequalities: Problems with Solutions
Irrational equations and Inequalities: Problems with Solutions

In this article, we will cover the concepts of irrational inequalities and equations. This concept falls under the broader category of complex numbers., a crucial Chapter in class 11 Mathematics. It is not only essential for board exams but also for competitive exams like the Joint Entrance Examination (JEE Main), and other entrance exams such as SRMJEE, BITSAT, WBJEE, BCECE, and more. Over the last ten years of the JEE Main exam (from 2013 to 2023), a total of three questions have been asked on this concept, including one in 2016, one in 2018, and one in 2019.

Inequalities

Inequalities are the relationship between two expressions that are not equal to one another. Symbols denoting the inequalities are <, >, ≤, ≥, and ≠.

  • $x<4$, "is read as $x$ less than $4^{\prime \prime}, x \leq 4$, is read as $x$ less than or equal to $4^{\prime \prime}$.
  • Similarly $x>4$, "is read as $x$ greater than $4^{\circ}$ and $x \geq 4$, "is read as $x$ greater than or equal to 4 ".

The process of solving inequalities is the same as of equality but instead of equality symbol inequality symbol is used throughout the process.

Types of Inequalities

  • Linear Inequalities: Involve linear expressions.
    • Example: $2 x+3 \leq 7$
  • Quadratic Inequalities: Involve quadratic expressions.
    • Example: $x^2-4 x+3 \geq 0$
  • Polynomial Inequalities: Involve polynomials of degree greater than two.
    • Example: $x^3-2 x^2+x-5<0$
  • Rational Inequalities: Involve ratios of polynomials.
    • Example: $\frac{x+1}{x-3} \geq 2$
  • Absolute Value Inequalities: Involve absolute value expressions.
    • Example: $|x-2| \leq 5$
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A few rules that are different from equality rules

  • - If we multiply or divide both sides of the inequality by a negative number, then we reverse the inequality (reversing inequality means > gets converted to < and vice versa, and $\geq$ gets converted to $\leq$ and vice versa) (eg $4>3$ means $-4<-3$ )
  • If we cross multiply a negative quantity in an inequality, then we reverse the inequality (eg $3>-2$ means $-3 / 2<1$ )
  • If we cancel the minus sign from both sides of an inequality, then we reverse the inequality. (eg $-3>-4$ means $3<4$ )
  • As we usually do not know the sign of a variable term like $x,(x-2)$, etc, so we do not cross-multiply them, as we cannot decide if we have to reverse the sign of inequality or not.

We get a range of solutions while solving inequality which satisfies the inequality,

for e.g. a $>3$ gives us a range of solutions, means a ? $(3, \infty)$
Graphically inequalities can be shown as a region belonging to one side of the line or between lines, for example, inequality $-3<x \leq 5$ can be represented as below, a region belonging to $-3$ and $5$ are the region of possible $x$ including $45$ and excluding $-3$ .

inequalities


Irrational equation

An irrational equation is an equation where the variable is inside the radical or the variable is a base of power with fractional exponents.

For example,

$\sqrt{2 x-3}=4$

To solve any irrational equation, eliminate the radical by raising the term in the other side to a power. For the equation given in the form of $\sqrt[n]{f(x)}=g(x)$, where $n \in N$ and $f(x), g(x)$ are polynomial function, solve $f(x)=(g(x))^n$,
and check the values of $x$ obtained by putting it in original equation

For example,

given equation is $\sqrt{x^2-4 x+4}=x+1$
then, $x^2-4 x+4=(x+1)^2$
$\Rightarrow \mathrm{x}^2-4 \mathrm{x}+4=\mathrm{x}^2+2 \mathrm{x}+1$
$\Rightarrow 6 \mathrm{x}=3 \Rightarrow \mathrm{x}=\frac{1}{2}$
$\mathrm{x}=\frac{1}{2}$ satisfies the original equation
so $\mathrm{x}=1 / 2$ is the answer

Irrational Inequalities:

Irrational inequalities are inequalities with irrational equations. For example, $\sqrt{x^2+7} = 4$

If $n$ is odd

To solve inequations of the form $(f(x))^{1 / n}>g(x)$ or $(f(x))^{1 / n}<g(x)$, or $(f(x))^{1 / n}>(g(x))^{1 / n}$, raise both sides to the power $n$, and solve to get the answer.

If $n$ is even

1. To solve inequations of the form $(f(x))^{1 / n}>g(x)$,
a. LHS should be defined, so solve $f(x) \geq 0$
b. Now if $\mathrm{g}(\mathrm{x})<0$, then LHS will be greater than RHS for all such values
c. If $g(x) \geq 0$, then solve $f(x)>(g(x))^n$

In the end take intersection of a with (b union c)
2. To solve inequations of the form $(f(x))^{1 / n}<g(x)$,
a. LHS should be defined, so solve $f(x) \geq 0$
b. Now if $\mathrm{g}(\mathrm{x})<0$, then LHS will be not be lesser than RHS for all such values
c. If $g(x) \geq 0$, then solve $f(x)<(g(x))^n$

In the end take intersection of $a$ and $c$

For example,

$\begin{aligned} 3+\sqrt{3 x+1}=x & \Rightarrow \sqrt{3 x+1}=x-3 \\ & \Rightarrow 3 x+1=(x-3)^2 \\ & \Rightarrow 3 x+1=x^2-6 x+9 \\ & \Rightarrow 0=x^2-9 x+8 \\ & \Rightarrow x=8, x=1\end{aligned}$

Recommended Video Based on Irrational Equations and Inequalities:

Solved Examples Based On Irrational Inequalities and Equations:

Example 1: Solve the inequality $\sqrt{x+14}<x+2$

Solution:

$\sqrt{x+14}<x+2$

1. For LHS to be defined, $x+14 \geq 0$ means $x \geq-14$
2. When $x+2 \leqslant 0$, then RHS cannot be greater than LHS, so no answer from this case
3. When $x+2 \geq 0$, means when $x \geq-2$,

In this case, we can square both sides

$
\begin{aligned}
& x+14<(x+2)^2 \\
& x+14<x^2+4 x+4 \\
& x^2+3 x-10>0 \\
& (x+5)(x-2)>0 \\
& x<-5 \text { or } x>2
\end{aligned}
$


Taking intersection with $x \geq-2$, which equals $x>2$
Now, the answer is the intersection of (1) and (3), which is $x>2$

Example 2: Which of the options is correct for the inequality $\sqrt{-x^2+4 x-3}>6-2 x ?$

1) $(1,3)$
2) $\left(\frac{13}{5}, 3\right)$
3) $(3, \infty)$
4) $(-\infty, 3)$

Solution

1. LHS should be defined, so

$-x^2+4 x-3 \geq 0$

$\begin{aligned} & x^2-4 x+3 \leq 0 \\ & (x-1)(x-3) \leq 0 \\ & 1 \leq x \leq 3\end{aligned}$

2. When RHS < 0, then all these values will satisfy the inequation

$6-2 x<0$

$\Rightarrow x>2$

3. When RHS 0 (when $x \leq 3$ ), then we can square the inequation

$
\begin{aligned}
& -x^2+4 x-3>(6-2 x)^2 \\
& -x^2+4 x-3>36+4 x^2-24 x \\
& 5 x^2-28 x+39<0 \\
& (x-3)(5 x-13)<0 \\
& 13 / 5<x<3
\end{aligned}
$

Taking the intersection of this result with $x<3$, we get the interval of x , i.e

$13 / 5<x<3$

Hence, the answer is the option 2.

Example 3: If x is a solution of the equation , $\sqrt{2 x+1}-\sqrt{2 x-1}=1,\left(x \geqslant \frac{1}{2}\right)$, then $\sqrt{4 x^2-1}$ is equal to :

Solution:


$
\begin{aligned}
& \sqrt{2 x+1}-\sqrt{2 x-1}=1 \\
& \Rightarrow \sqrt{2 x+1}=\sqrt{2 x-1}+1
\end{aligned}
$

square both side

$
\begin{aligned}
& \Rightarrow 2 x+1=2 x-1+1+2 \sqrt{2 x-1} \\
& \Rightarrow 1=2 \sqrt{2 x-1}
\end{aligned}
$

square both side

$
\begin{aligned}
& \Rightarrow 2 x-1=\frac{1}{4} \\
& \Rightarrow x=\frac{5}{8}
\end{aligned}
$

Now $\sqrt{4 x^2-1}$ at $x=5 / 8 \Rightarrow \sqrt{4 \times \frac{25}{64}-1}=3 / 4$

Example 4: Let $S={ }^{-1}$ R: $x>0$ 0 and $2|\sqrt{x}-3|+\sqrt{x}(\sqrt{x}-6)+6=0$ Then S:

1) contains exactly four elements

2) is an empty set.

3) contains exactly one element

4) contains exactly two elements

Solution:

As we learned in

Roots of Quadratic Equation -

$\alpha=\frac{-b+\sqrt{b^2-4 a c}}{2 a}$

$\beta=\frac{-b-\sqrt{b^2-4 a c}}{2 a}$

- wherein

$a x^2+b x+c=0$

is the equation

$a, b, c \in R, \quad a \neq 0$

Case 1

$\sqrt{x} \geq 3 \Rightarrow x \geq 9$

$2(t-3)+t(t-6)+6=0$

$t^2-4 t=0$

$\Rightarrow t=0, t=4$

$\sqrt{x}=0, \sqrt{x}=4$

$x=0, x=16$

we take $x=16$ $x \geq 9$

case 2

$0<\sqrt{x}<3 \Rightarrow 0<x<9$

$-2 t+6+t^2-6 t+6=0$

$t^2-8 t+12=0$

$\Rightarrow t=2, t=6$

$\Rightarrow x=4, x=36$

Thus $x=4$ : $x<9$

So there are two elements

Hence, the answer is the option 4.

Example 5: Solve $x-3 \sqrt{x+1}+3=0$

Solution:

$\begin{aligned} & x-3 \sqrt{x+1}+3=0 \\ & \text { Let } \sqrt{x+1}=t \\ & \Rightarrow x+1=t^2 \\ & \Rightarrow x=t^2-1\end{aligned}$

So, the equation becomes

$\begin{aligned} & \left(t^2-1\right)-3 t+3=0 \\ \Rightarrow & t^2-3 t+2=0 \\ \Rightarrow & (t-1)(t-2)=0 \\ \Rightarrow & t=1, t=2 \\ \Rightarrow & \sqrt{x+1}=1, \quad \sqrt{x+1}=2 \\ \Rightarrow & x+1=1, \quad x+1=4 \\ \Rightarrow & x=0, \quad x=3\end{aligned}$


Frequently Asked Questions (FAQs)

1. What are inequalities?

Inequalities are the relationship between two expressions that are not equal to one another.

2. What are irrational equations?

An irrational equation is an equation where the variable is inside the radical or the variable is a base of power with fractional exponents.

3. All the pairs (x,y) that satisfy the inequality $2^{\sqrt{\sin ^2 x-2 \sin x+5}} \cdot \frac{1}{4^{\sin ^2 y}} \leq 1$ also satisfy the equation:

$2^{\sqrt{\sin ^2 x-2 \sin x+5}} \cdot \frac{1}{4^{\sin ^2 y}} \leq 1$

$ 2^{\sqrt{\sin ^2 x-2 \sin x+5}<2^{2 \sin 2}}$

$\sqrt{\sin ^2 x-2 \sin x+5} \leq 2 \sin ^2 y$

$\sqrt{(\sin x-1)^2+4} \leq 2 \sin ^2 y$

$\Rightarrow \sin x=1 \&|\sin y|=1$

4. Solve $-5(x-1) \leq 10(2 x-3)$

$-5(x-1) \leqslant 010(2 x-3)$

$\Rightarrow \quad(x-1) \geqslant \frac{10}{-5}(2 x-3)$

$\Rightarrow x-1 \geqslant-2(2 x-3)$

$\Rightarrow x-1 \geqslant-4 x+6$

$\Rightarrow x+4 x \geqslant 6+1$

$\Rightarrow 5 x \geqslant 7$ 

$\Rightarrow x \geqslant \frac{7}{5}$

5. If $\frac{-1}{2}<x \leq 3$ then $\frac{1}{x} \epsilon$.

 $-\frac{1}{2}<x \leqslant 3\left(-\frac{1}{2}<0,3>0\right)$

$\Rightarrow-\frac{1}{2}<x<0^{-} \text {or } 0^{+}<x \leqslant 3$

$\Rightarrow-2>\frac{1}{x}>-\infty \text { or } \infty>\frac{1}{x} \geqslant \frac{1}{3}$

$\Rightarrow \frac{1}{x} \in(-\infty,-2) \cup\left[\frac{1}{3}, \infty\right)$

6. What is an irrational equation?
An irrational equation is an equation that contains at least one term with a variable under a square root (or other root) sign. These equations often require special techniques to solve, as the presence of the root can complicate the process.
7. How do irrational inequalities differ from irrational equations?
Irrational inequalities are similar to irrational equations but use inequality symbols (<, >, ≤, ≥) instead of an equals sign. They require additional steps to solve, including considering the domain of the expression and potential extraneous solutions.
8. Why is it important to check solutions when solving irrational equations?
Checking solutions is crucial because the process of solving irrational equations often involves squaring both sides, which can introduce extraneous solutions. These are values that satisfy the squared equation but not the original equation.
9. What is the domain of an irrational equation, and why is it important?
The domain of an irrational equation is the set of all possible values for the variable that make the equation meaningful. It's important because it helps identify restrictions on the solution set and can prevent errors when solving the equation.
10. How do you solve an equation containing a square root?
To solve an equation with a square root, typically you isolate the square root term on one side, then square both sides of the equation. This eliminates the square root, but remember to check for extraneous solutions afterward.
11. What is the difference between rational and irrational equations?
Rational equations contain variables in the numerator or denominator of fractions, while irrational equations contain variables under root signs. Irrational equations often require more complex solving techniques and careful consideration of domains.
12. Can an irrational equation have no solution? If so, when?
Yes, an irrational equation can have no solution. This occurs when the domain restrictions of the irrational expression cannot be satisfied by any real number, or when squaring both sides leads to a contradiction.
13. What is meant by "rationalizing" in the context of irrational equations?
Rationalizing refers to the process of eliminating irrational terms (like square roots) from an equation. This is often done by multiplying both sides of the equation by a carefully chosen expression that cancels out the irrational term.
14. How do you solve an irrational inequality?
To solve an irrational inequality: 1) Isolate the irrational term, 2) Square both sides if dealing with a square root, 3) Solve the resulting rational inequality, 4) Check the domain of the original inequality, and 5) Test points to determine the final solution set.
15. Why might squaring both sides of an irrational equation change the solution set?
Squaring both sides can change the solution set because it's not a reversible operation. It can introduce extraneous solutions that satisfy the squared equation but not the original one. This is why checking solutions is crucial.
16. What is the significance of the discriminant in solving irrational equations involving quadratic expressions?
The discriminant (b²-4ac in a quadratic ax²+bx+c) helps determine the nature of the solutions to a quadratic equation. In irrational equations, it can indicate whether the equation has real solutions, which is crucial for determining the overall solution set.
17. How do you handle an irrational equation with multiple radical terms?
For equations with multiple radical terms, isolate one radical term on one side of the equation. Then, square both sides to eliminate that radical. Repeat this process until all radicals are eliminated, being careful to check for extraneous solutions at each step.
18. What role does graphing play in understanding irrational equations and inequalities?
Graphing can provide visual insight into the behavior of irrational equations and inequalities. It can help identify the number and approximate location of solutions, as well as illustrate the domain and range of the expressions involved.
19. How does the concept of absolute value relate to irrational equations?
Absolute value can appear in irrational equations, often in the form of square roots of squared terms. Understanding that √(x²) = |x| is crucial for solving these types of equations and interpreting their solutions correctly.
20. What is the relationship between complex numbers and irrational equations?
Some irrational equations may lead to complex number solutions, especially when the equation involves even-indexed roots of negative numbers. Understanding complex numbers is essential for fully characterizing the solution set of certain irrational equations.
21. How do you determine if an irrational equation will have real solutions?
To determine if an irrational equation has real solutions, consider the domain of the irrational expressions and any restrictions imposed by the equation. If the domain is empty or the restrictions cannot be satisfied by real numbers, the equation has no real solutions.
22. What is the significance of the term under the radical in an irrational equation?
The term under the radical is crucial as it determines the domain of the equation. For square roots, this term must be non-negative for real solutions. Understanding this helps in solving the equation and interpreting its solutions.
23. How do you solve an irrational equation involving cube roots?
For cube root equations, isolate the cube root term, then cube both sides of the equation. Unlike with square roots, cubing doesn't introduce extraneous solutions, but it's still important to check the domain of the original equation.
24. What are some common mistakes students make when solving irrational equations?
Common mistakes include: forgetting to check for extraneous solutions, ignoring domain restrictions, incorrectly applying algebraic operations to radicals, and failing to consider the possibility of complex solutions.
25. How do you interpret the solution to an irrational inequality graphically?
Graphically, the solution to an irrational inequality represents the x-values where the graph of one side of the inequality is above or below (depending on the inequality sign) the graph of the other side.
26. What is the connection between irrational equations and the concept of exponents?
Irrational equations often involve roots, which are fractional exponents. Understanding the properties of exponents, especially fractional and negative exponents, is crucial for manipulating and solving irrational equations.
27. How do you approach an irrational equation where the variable appears both inside and outside the radical?
For such equations, try to isolate terms with the variable inside the radical on one side and terms with the variable outside on the other. Then square both sides. This may lead to a quadratic equation, which can be solved using standard methods.
28. What is the importance of understanding function composition in solving irrational equations?
Function composition is often implicitly used in solving irrational equations, especially when isolating radical terms. Understanding how composite functions work helps in correctly manipulating the equation and interpreting the results.
29. How do you solve an irrational inequality involving absolute values?
To solve an irrational inequality with absolute values, first solve the equation formed by replacing the inequality sign with an equals sign. Then, use the solutions as boundary points to test intervals and determine where the inequality holds true.
30. What is the role of the zero-product property in solving some irrational equations?
The zero-product property (if ab = 0, then a = 0 or b = 0) is often useful after squaring both sides of an irrational equation. It helps in breaking down the resulting equation into simpler parts that can be solved individually.
31. How do you determine the number of solutions an irrational equation might have?
The number of solutions depends on the specific equation. Consider the degree of the equation after rationalizing, the nature of the irrational terms, and any domain restrictions. Graphing can also provide insight into the number of intersections or solutions.
32. What is the significance of the principle of "corresponding parts of congruent equations" in solving irrational equations?
This principle states that if two expressions are equal, their corresponding parts must also be equal. It's useful in solving complex irrational equations by allowing you to equate the rational and irrational parts separately.
33. How do you approach an irrational equation that results in a higher-degree polynomial after squaring?
If squaring leads to a higher-degree polynomial, standard techniques like factoring, using the quadratic formula (for degree 2), or polynomial solving methods (for higher degrees) may be necessary. Always check solutions in the original equation.
34. What role does interval notation play in expressing solutions to irrational inequalities?
Interval notation is a concise way to express the solution set of irrational inequalities. It clearly shows the range of values that satisfy the inequality, including any restrictions or discontinuities in the solution set.
35. How do you solve a system of irrational equations?
To solve a system of irrational equations, you can use substitution or elimination methods, but be careful to square terms appropriately. After solving, check solutions in both original equations to eliminate any extraneous solutions introduced by squaring.
36. What is the relationship between irrational equations and the concept of inverse functions?
Inverse functions are often used implicitly when solving irrational equations. For example, squaring both sides of an equation involving a square root is equivalent to applying the inverse function (squaring) to both sides.
37. How do you approach an irrational equation where the variable appears in multiple radicals?
For equations with multiple radicals, isolate one radical term and square both sides. Repeat this process until all radicals are eliminated. Be extra careful to check for extraneous solutions, as each squaring step can introduce them.
38. What is the importance of understanding the behavior of even and odd functions in solving irrational equations?
Understanding even and odd functions helps in predicting the behavior of solutions. For example, even-indexed roots of negative numbers may lead to complex solutions, while odd-indexed roots always have real solutions for real inputs.
39. How do you solve an irrational equation that involves logarithms?
For equations involving both radicals and logarithms, try to isolate the radical term first, then square both sides. This often results in an equation with logarithms, which can be solved using logarithm properties and exponential functions.
40. What is the significance of the method of completing the square in solving some irrational equations?
Completing the square can be useful in irrational equations, especially when squaring both sides results in a quadratic equation. It helps in simplifying the equation and can make it easier to identify the nature and number of solutions.
41. How do you determine if an irrational inequality is strict or non-strict?
The strictness of an irrational inequality depends on the original inequality sign (< or > for strict, ≤ or ≥ for non-strict) and the nature of the irrational expression. Be careful when squaring, as it can affect the inequality direction for negative terms.
42. What is the role of parametric equations in solving certain types of irrational equations?
Parametric equations can be useful in solving complex irrational equations by introducing a parameter to simplify the equation. This approach can help in finding solutions that might be difficult to obtain through direct algebraic manipulation.
43. How do you approach an irrational equation that results in a rational equation with a denominator of zero?
If solving an irrational equation leads to a rational equation with a zero denominator, it usually indicates that the original equation has no solution for that particular value. Always check the domain of the original equation to confirm this.
44. What is the importance of understanding the concept of continuity in solving irrational inequalities?
Continuity is crucial in solving irrational inequalities because it helps in identifying where the inequality might change direction. Points of discontinuity often occur at the boundaries of the domain and can be critical points in determining the solution set.
45. How do you solve an irrational equation involving nested radicals?
For nested radicals, start by isolating the outermost radical. Square both sides to remove it, then repeat the process for inner radicals. This often leads to a complex equation, so be extra vigilant about checking for extraneous solutions.
46. What is the significance of the intermediate value theorem in understanding solutions to irrational equations?
The intermediate value theorem helps in proving the existence of solutions to irrational equations. If a continuous function changes sign over an interval, this theorem guarantees that there's at least one solution within that interval.
47. How do you approach an irrational equation where the variable appears in the base and the exponent?
For equations where the variable is both in the base and exponent, try to isolate terms with the variable in the exponent on one side. Then use logarithms to bring the variable down from the exponent. This often results in an equation that can be solved algebraically.
48. What is the role of asymptotes in solving irrational inequalities?
Asymptotes are important in irrational inequalities as they often represent boundaries of the solution set. Vertical asymptotes can indicate where the inequality changes direction, while horizontal asymptotes can help in understanding the behavior of the inequality for large values of the variable.
49. How do you solve an irrational equation that involves trigonometric functions?
For equations with both radicals and trigonometric functions, try to isolate the radical term first. After squaring, you'll likely have an equation involving trigonometric functions, which can be solved using trigonometric identities and techniques.
50. What is the significance of the rational root theorem in solving some irrational equations?
The rational root theorem can be useful after rationalizing an irrational equation, especially if it results in a polynomial equation. It helps in identifying potential rational solutions, which can simplify the solving process.
51. How do you approach an irrational inequality that involves piecewise functions?
For piecewise irrational inequalities, solve the inequality for each piece separately, paying attention to the domain of each piece. Then combine the results, being careful about the intervals where each piece is defined.
52. What is the importance of understanding the concept of monotonicity in solving irrational inequalities?
Monotonicity helps in determining how an irrational inequality behaves over different intervals. If a function is monotonic (always increasing or always decreasing) over an interval, it simplifies the process of determining where the inequality is satisfied.
53. How do you solve an irrational equation that involves complex numbers?
For irrational equations with complex numbers, use similar techniques as with real numbers, but be prepared for complex solutions. Remember that even-indexed roots of negative numbers will yield complex results. Always express final answers in standard complex form (a + bi).
54. What is the role of the squeeze theorem in understanding solutions to some irrational inequalities?
The squeeze theorem can be useful in solving complex irrational inequalities, especially when direct algebraic methods are difficult. It helps in determining the behavior of the inequality by comparing it to simpler functions that bound it from above and below.
55. How do you approach an irrational equation or inequality that seems to have no algebraic solution?
For equations or inequalities without clear algebraic solutions, consider numerical methods or graphical approaches. Plotting the functions involved can provide insight into the number and approximate location of solutions. In some cases, computer algebra systems may be necessary for more precise results.

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