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How To Solve Equations With Radicals

The first step to solve radical equation is to isolate the radical on the left hand side of the equation. After that think of an operation that can eliminate the radical notation. Let see some simple example.

Solve $\sqrt{x+5}+2=4$

Add $-2$ to both sides to eliminate the 2 from the left to get $\sqrt{x+5}=4$ . Now the operation that can eliminate the square root is squaring. So squaring both sides, we have $x+5= 4$ . Hence $x=-1$ . Now we need to check that this result is really a solution by plugin back to the original equation. For this problem one can easily check that $x=-1$ is a solution.

Why we need to plugin back the final result to the original equation? The reason of this is that the process of squaring (or raising to even power) doesn't give us an equivalent equation. To illustrate this onsider $x=5$ if we square both sides we have $x^2=25$ . But the last equation has two solutions $x=\pm 5$ .

The videos below will give you examples how to solve many kind of equations with radicals.

How To Solve Quadratic Inequalities

The way to solve quadratic inequalities is similar to solve quadratic equation. The first step is to move all the term to one side of the inequality say to the left to get something lik $px^2+qx+r<0$ or $px^2+qx+r>0$ and then try to factor the left hand side intos something of the form $p(x+a)(x+b)$ . Now to show that this expression is negative (or positive) we need to do some analysis on where the value of $x+a$ and $x+b$ are positive or negative. To make the analysis easier we use the real line to help us to determine this.

To make everything clear, you can watch the videos below

How To Solve Linear Inequalities

The way to solve linear inequalities is similar to solving linear equation. You want to bring and isolate the $x$ variable into one side of the inequality. You may do the following to the inequalities

Add/Subtract the same amount of number to both sides of the inequality
Multiply/divide both sides of the inequality by a positive number without changing the inequality
Multiply/divide both sides of the inequality by a negative number and reverse the inequality sign

By using the above operation repeatedly we can bring $x$ and isolate it to one side of the inequality.

The videos below will give you an example how to do these operations to solve the inequality.