Solving Radical Inequalities

Any algebraic expressions or equations may contain **inequalities** to show an **order relationship** between them. Here, we will discuss inequalities in radical functions (containing square roots) and learn how to solve them.

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Jetzt kostenlos anmeldenAny algebraic expressions or equations may contain **inequalities** to show an **order relationship** between them. Here, we will discuss inequalities in radical functions (containing square roots) and learn how to solve them.

An inequality that has variables within the radicand is known as ** radical inequality**.

In other words, a radical inequality is an inequality that has a variable or variables inside the radical symbol (the radicand). The following formula describes the format in which you can expect to see radical inequalities:$\sqrt[n]{x}<d$. The variable x inside the radical represents the radicand. This format is the same for all of the other inequality signs$>,\le ,\ge .$

Remember that radicand is a value inside the radical symbol. That is, it is the value for which we take the root.

Let us take a look at some examples to understand how we can **identify** this inequality.

$2\sqrt{x}-5\ge 3$.

Here we see a variable x inside the root. And this equation is expressed with an inequality sign.

Similarly, below are some other examples of radical inequalities.

$4\sqrt{x+1}\ge 12\phantom{\rule{0ex}{0ex}}\sqrt{x+3}-2\le 9\phantom{\rule{0ex}{0ex}}\sqrt[3]{x-4}<3\phantom{\rule{0ex}{0ex}}\frac{1}{6}{\left(12a\right)}^{\frac{1}{3}}>1$

The following are the two ways to determine radical inequalities:

- Using algebra
- Using graphs

We can solve radical inequalities of all types by using algebra. Some radical inequalities also have variables outside the radical, and we can use algebra to calculate those as well. The following steps can be used to solve radical inequalities:

** Step 1**: Check the index of the radical.

- If the index is even, then the final calculated value of the radicand cannot be negative and must be positive. This is called a
**domain restriction**. Here for the radical inequality,$\sqrt[n]{x}<d$, n is the index and x is the radicand.

** Step 2**: If the index is even, consider the radicand value as positive. Solve for the variable x within the radicand.

- So, we will solve the variable x for this radicand when it is greater than or equal to zero. That is, we consider the radicand as$x\ge 0$from the radical inequality$\sqrt[n]{x}<d$and calculate the variable x. If the index is odd, however, then consider the radicand as $x<d$.

As the principal square root is never negative, the inequalities which simplify to the form $\sqrt{x}\le d$, where d is the negative number, have no solutions.

** Step 3**: Solve the

- We remove the radical by taking the index and using it as an exponent on the terms on both sides of the inequality. (i.e., $\sqrt[n]{x}<d\Rightarrow {\left(\sqrt[n]{x}\right)}^{n}<{d}^{n}$). Here note that when using the index as an exponent on the radical term, it nullifies the radical symbol, therefore eliminating it.

Remember that when no index value is given then it is always considered as 2.

** Step 4**: Test the values to check the solution.

- To test the values of x we will consider some random values which satisfy the inequality. And we will also consider some values outside equality so that we can confirm the correctness of our solution.

Let us consider an example to properly understand it.

Solve $2+\sqrt{4x-4}\le 6$

__Solution__: To solve this radical inequality, let's follow all the steps.

Step 1: First we check the index of the given radical inequality. As no index value is given, the index value is 2.

Step 2: As the index is even, the radicand of the square root will be greater than or equal to zero.

$4x-4\ge 0\phantom{\rule{0ex}{0ex}}\Rightarrow 4x\ge 4\phantom{\rule{0ex}{0ex}}\Rightarrow x\ge 1\left(1\right)$

Step 3: Now we will solve the radical inequality algebraically and also remove the radical symbol to simplify it. First, we isolate the radical.

$2+\sqrt{4x-4}\le 6\Rightarrow \sqrt{4x-4}\le 4$

Now, we eliminate the radical symbol by taking index as an exponent on both sides of the inequality.

${\left(\sqrt{4x-4}\right)}^{2}\le {4}^{2}\phantom{\rule{0ex}{0ex}}\Rightarrow 4x-4\le 16\phantom{\rule{0ex}{0ex}}\Rightarrow 4x\le 20\phantom{\rule{0ex}{0ex}}\Rightarrow x\le 5\left(2\right)$

Here, we got two inequalities for the value of x from equation $\left(1\right)$ and $\left(2\right)$. So we combine them both and write it as a compound inequality. So our final answer is:

$1\le x\le 5$

Here note that 1 is the lower interval value and 5 is the upper interval value.

Step 4: Finally, we will test some values of x to check our solution and confirm it. Let's consider $x=0,8$ outside our range of x and $x=3$ inside our range of x.

$x=0$ | $x=3$ | $x=8$ |

$2+\sqrt{4\left(0\right)-4}\phantom{\rule{0ex}{0ex}}2+\sqrt{-4}$Not possible as $\sqrt{-4}$ is not a real number | $2+\sqrt{4\left(3\right)-4}\phantom{\rule{0ex}{0ex}}2+\sqrt{8}\phantom{\rule{0ex}{0ex}}\approx 4.8<6$It satisfies the inequality | $2+\sqrt{4\left(8\right)-4}\phantom{\rule{0ex}{0ex}}2+\sqrt{28}\phantom{\rule{0ex}{0ex}}\approx 7.3\nleqq 6$This value doesn't satisfy the inequality |

We can see that the value of x satisfies the radical inequality. So the solution $\mathbf{1}\mathbf{\le}\mathit{x}\mathbf{\le}\mathbf{5}$ checks and satisfies the given radical inequality.

We can also solve radical inequalities with the help of graphs. We will follow the given steps to use this method:

** Step 1**: For a radical inequality $\sqrt[n]{x}<d$, where $>$could also be$>,\le ,\ge $, consider both functions of $y=\sqrt[n]{x}$ and $y=d$.

__ Step 2__: Plot a graph which displays both of the functions from step 1.

** Step 3**: Identify the interval(s) of x for the graphed functions by comparing them graphically, taking the inequality sign into account. Also, find the point x where both functions intersect on the graph, if applicable.

- That is, if the inequality has a greater than sign, then find values of x
**above**the other function. And if the inequality has a less than sign, then identify x**below**the other function.

** Step 4**: Confirm and test the values of x.

- Similarly to the previous method, we consider random values of x which satisfy the inequality, and also which do not satisfy the obtained range of x.

Let us understand this with the help of an example.

Solve $\sqrt{x-5}>3$

__Solution__:

Step 1: We consider $y=\sqrt{x-5}$ and $y=3$.

Step 2: Now we plot the graph for both the functions from step 1.

Here, the graph with the red line is of the function $y=\sqrt{x-5}$and the graph with the green line is of the function $y=3$. We have plotted the graph such that we can clearly identify the x-axis values between 0 and 30. Similarly, we can clearly take note of the values on the y-axis.

Step 3: Now, we identify the values of x for which the first graph $y=\sqrt{x-5}$ lies above the second graph $y=3$, as we have a greater than inequality sign in the original inequality equation. We can also see that value of x intersects both the graphs at $x=14$. This implies that the first graph lies above the second graph for $x>14.$

$\therefore $The solution to this radical inequality is $\mathit{x}\mathbf{>}\mathbf{14}$.

Note: The domain of ${y}{=}\sqrt{x-5}$ is ${x}{\ge}{5}$. So the domain does not have any effect on the solution.

Here, some other examples are presented of radical inequality using both methods.

Solve $\sqrt[3]{x+2}\ge 1,$ algebraically and graphically.

__Solution__: First we solve using the algebra method.

Step 1: First we check the index of the given radical inequality. Here it is 3.

Step 2: As the index is odd, we consider: $x+2\ge 1\Rightarrow x\ge -1.$

Step 3: Now we solve the original inequality. As the given inequality does not have other operations, we skip the step of isolation.

$\sqrt[3]{x+2}\ge 1\phantom{\rule{0ex}{0ex}}\Rightarrow {\left(\sqrt[3]{x+2}\right)}^{3}\ge {1}^{3}\phantom{\rule{0ex}{0ex}}\Rightarrow x+2\ge 1\phantom{\rule{0ex}{0ex}}\therefore x\ge -1$

So the values of x from step 2 and step 3 are $x\ge -1$.

Step 4: Now we check our solution to confirm it.

$x=-2$ | $x=-1$ | $x=12$ |

$\sqrt[3]{-2+2}\phantom{\rule{0ex}{0ex}}=\sqrt[3]{0}\phantom{\rule{0ex}{0ex}}=0\ngeqq 1$ | $\sqrt[3]{-1+2}\phantom{\rule{0ex}{0ex}}=\sqrt[3]{1}\phantom{\rule{0ex}{0ex}}=1$ | $\sqrt[3]{12+2}\phantom{\rule{0ex}{0ex}}=\sqrt[3]{14}\phantom{\rule{0ex}{0ex}}\approx 2.4\ge 1$ |

Hence, it can be seen that the given radical inequality is satisfied for the values$\mathit{x}\mathbf{\ge}\mathbf{-}\mathbf{1}$.

Now we will solve the same radical inequality using the graphical method.

Step 1: We consider $y=\sqrt[3]{x+2}$ and $y=1.$

Step 2: we plot the graphs for two functions taken into account in step 1.

In the above graph, the red line represents the function $y=\sqrt[3]{x+2}$ and the blue line represents the function$y=1$.

Step 3: We identify the values of x for which the first graph $y=\sqrt[3]{x+2}$ lies above the graph $y=1$. And the value of x intersects both the graphs at $x=-1$. So the first graph lies above the second graph for the values $x\ge -1$.

Hence, the solution for the given radical inequality is $\mathit{x}\mathbf{\ge}\mathbf{-}\mathbf{1}$.

- An inequality that has variables within the radicand is known as radical inequality.
- The radicand is the value inside of the radical symbol.
- There are two ways to determine radical inequalities: using algebra and using graphs.

An inequality that has variables within the radicand is known as radical inequality.

Radical inequalities can be solved using algebra and graphs.

Radical inequalities can be graphed by plotting both the expression on the sides of inequality sign.

What are radical inequalities?

Radical inequalities are inequalities that have variables within the radicand.

By which of the following methods radical inequalities can be solved?

Algebra

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