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How To Solve Rational Inequalities

The idea to solve rational inequalities is first to rewrite the inequality so that the left hand side is a single rational function $\frac{p(x)}{q(x)}$ and the right hand side is zero. So the firs step is to rewrite the inequality so it looks like $\frac{p(x)}{q(x)}<0$ (of course here the inequality symbol can be $>,\leq$ or $\geq$ ). The next step is to find for which $x$ the numerator is zero and for which $x$ the denominator is zero. Plot all these zeros on a number line. These zeros will divide the number line into several subintervals. Take a sample point on each sub interval and plugin to $\frac{p(x)}{q(x)}>0$ . If you get a right statement after you plugin your $x$ to the inequality, that means the subinterval is a solution to the inequality.

Another method is that after you rewrite your inequality so that it has the form $\frac{p(x)}{q(x)}<0$ , then you consider two cases. the case when $q(x)<0$ and the case when $q(x)>0$ . If $q(x)<0$ then multiplying the inequality by $q(x)$ we have $p(x)>0$ (remember we need to reverse the inequality sign since $q(x)$ is negative) and then solve the inequality $p(x)>0$ . If $q(x)>0$ , multiplying both sides of the inequality by $q(x)$ we have $p(x)<0$ and then we can solve this inequality.

The video belows will guide you to use these to methods to solve rational inequalities.

Solving Absolute Value Inequality

Solving absolute value inequalities is based on these two properties of absolute value: We have $|x|a$ equivalent to $x<-a$ or $x>a$ . So the strategy of solving absolute value inequalities is to bring the term with absolute value to the left of the inequality and put the other terms on the right and then use the above property to remove the absolute value sign and solve the inequality.The videos below will give you a detail expanation about this and do some couple of problems using this principle. >

How To Solve Absolute Value Equation

What is the $x$ that makes $|x|=2$ ? The answer is really simple $x=2$ or $x=-2$ . In general if $a>0$ (this is important). The equation $|x|=a$ always has to solutions, namely $x=\pm a$ . So the strategy of solving equation with absolute value is to bring the absolute value to the left hand side and the rest on the right hand side and then use the property that $|x|=a$ has solutions $x=\pm a$ . Here is an example:

Solve $|2x+1|+5=7$ .

Rewrite the equation as $|2x+1|=2$ . Now think of the terms inside the absolute value as one single entity. Then using the property mentioned above this entity must be equal to 2 or -2, mathematically $2x+1=2$ or $2x+1=-2$ and then you can solve each of these equation.

The video playlist below will give you more examples on how to solve equation with absolute value.