High School Math Solutions – Inequalities Calculator, Logarithmic Inequalities

Last post, we talked about radical inequalities. In this post, we will talk about how to solve logarithmic inequalities. We’ll see logarithmic inequalities in forms such as \log_b(f(x))<a or \ln(⁡f(x))<a. In order to solve these inequalities, the goal will be to isolate the variable, just as in any inequality, and we will do this by getting rid of the log function. Let’s dive in and see how to solve logarithmic inequalities.

Steps to solve logarithmic inequalities:
1. Use algebraic manipulation to move anything that is not in the logarithmic expression to one side
2. Combine logarithmic expressions
3. Isolate the variable by getting rid of the logarithmic expression
             Ex:    \log_b⁡(f(x))<a

b^(\log_b⁡(f(x))) <b^a

f(x)<b^a

Ex: \ln(f(x))<a

\ln⁡(f(x))<\ln⁡(e^a )

f(x)<e^a
4. Solve inequality
5. Get the range for the expression in the original log function
             Ex:  \log_b⁡(f(x))
Range: f(x)>0
6. Combine ranges

Let’s do an example step by step now.

First example (click here):

                                            \log_4⁡(x+3)-\log_4⁡(x+2)\ge\frac{3}{2}

Step 1: Use algebraic manipulation to move anything that is not in the logarithmic expression to one side

There’s nothing to move, so we can skip this step.

Step 2: Combine logarithmic expressions

                                            \log_4⁡(x+3)-\log_4⁡(x+2)\ge\frac{3}{2}

                                                      \log_4⁡(\frac{x+3}{x+2})≥\frac{3}{2}


Step 3: Isolate the variable by getting rid of the logarithmic expression

                                                 \log_4⁡(\frac{x+3}{x+2})≥\frac{3}{2}

                                                4^(\log_4⁡(\frac{x+3}{x+2})) ≥4^(\frac{3}{2})

                                                         \frac{x+3}{x+2}≥8

Step 4: Solve inequality

                                                          \frac{x+3}{x+2}≥8

We can see that this is now a rational inequality. We won’t solve this step by step; I will show the answer after solving this inequality. If you are struggling with solving this inequality, visit the blog post on rational inequalities.

                                                             -2<x≤\frac{-13}{7}

Step 5: Get the range for the expression in the original log function

                                                   \log_4⁡(x+3)-\log_4⁡(x+2)\ge\frac{3}{2}

                                                       x+3>0              x+2>0

                                                       x>-3              x>-2

Step 6: Combine ranges

                                                    -2<x≤\frac{-13}{7},    x>-2,    x>-3

                                                              -2<x≤\frac{-13}{7}

That wasn’t too bad! Let’s see some more examples.


Second example (click here):

Last example (click here):

Solving logarithmic inequalities is not too difficult. Just remember to get the ranges inside the logarithmic expressions and to double check your work. For more help and practice on this topic visit Symbolab’s  practice.

Until next time,

Leah

High School Math Solutions – Inequalities Calculator, Radical Inequalities

Last post, we went over how to solve absolute value inequalities. For today’s post, we will talk about how to solve radical inequalities. Solving radical inequalities is easier than solving absolute value inequalities and require fewer steps. Let’s see the steps on how to solve these inequalities.

Steps:

1. Isolate the square root
2. Check that the inequality is true (i.e. not less than 0)
3. Find the real region for the square root, (i.e. see when the expression inside square root is greater than or equal to 0)
4. Simplify and compute the inequality
5. Combine the ranges

Let’s see how to do one example step by step.

First example (click here):

                                                                    \sqrt{5+x}-1<3

Step 1: Isolate the square root

                                                                    \sqrt{5+x}<4

Step 2: Check that the inequality is true

Yes, this inequality is true the radical is not less than 0.

Step 3: Find the real region for the square root

                                                                       5+x≥0

                                                                         x≥-5

Step 4: Simplify and compute the inequality

                                                                    \sqrt{5+x}<4

                                                                (\sqrt{5+x})^2<4^2

                                                                     5+x<16

                                                                       x<11
Step 5: Combine the ranges

                                                                  x≥-5 and x<11

                                                                     -5≤x<11

That wasn’t too difficult. Let’s see some more examples.

Second example (click here):

Last example (click here):

Solving radical inequalities isn’t too difficult, however, they require practice. For more practice examples check out Symbolab’s practice.

Until next time,

Leah

High School Math Solutions – Polynomials Calculator, Dividing Polynomials (Long Division)

Last post, we talked dividing polynomials using factoring and splitting up the fraction. In this post, we will talk about another method for dividing polynomials, long division. Long division with polynomials is similar to the basic numerical long division, except we are dividing variables. This is where it gets tricky. I will talk about the steps to dividing polynomials using long division to help make the process easier and go into detail.

Steps for polynomial long division:

1.  Organize each polynomial by higher order

• We want to make sure that each polynomial is written in order of the variable with the highest exponent to the variable with the lowest exponent

• you can skip this step if they are already in high order

2.  Set up in long division form

• The denominator becomes the divisor and the numerator becomes the dividend

3.  Write 0 as the coefficient for missing terms in the dividend

• Since we’ve put in order the terms based on their exponent, we can see which terms are missing (i.e. x^4+x^2 we can see we are missing an x^3 term so we will add that in to its proper spot and make the coefficient 0) 

• This will help you with step 6, so you don’t subtract the wrong terms 

• You can skip this step if there are no missing terms

4.   Divide the first term of the dividend (numerator) by the first term of the divisor (denominator)

• This is allows us to see what we need to multiply the divisor by to get rid of the first term of the dividend

5.   Multiply the divisor by that term

•  Write the term down on top of line where the term that is getting eliminated is 

6.  Subtract this from the dividend 

• This gives you a new polynomial to work with

7.  Repeat steps 4-6 until you get a remainder 

• When you repeat step 4, move onto the newest first term from step 6

8.  Put the remainder over the divisor to create a fraction and add it to the new polynomial

This may seem a bit confusing, so we will go through two examples step by step to understand better how to solve these problems.

First example (click here):
                                                         \frac{(x^4+6x^2+2)}{(x^2+5)}

1.  Organize each polynomial by high order
     We can skip this step because the polynomials are already in high order
2.  Set up in long division form

3.  Write 0 as the coefficient for missing terms in the dividend

4.  Divide the first term of the dividend (numerator) by the first term of the divisor (denominator)
                                                   \frac{x^4}{x^2}= x^2

5.  Multiply the divisor by that term
     x^2∙(x^2+5)=x^4+5x^2

6.  Subtract this from the dividend

7.  Repeat step 4-6 until you get a remainder
     \frac{x^2}{x^2} =1

8.  Put the remainder over the divisor to create a fraction and add it to the new polynomial
                                                              x^2+1+\frac{(-3)}{(x^2+5)}

 Last example (click here):
                                                             \frac{(2x^2-18+5x)}{(x+4)}

1.  Organize each polynomial by high order
                                                              \frac{(2x^2+5x-18)}{(x+4)}

2.  Set up in long division form

3.  Write 0 as the coefficient for missing terms in the dividend
     We can skip this step because there are no missing terms.

4.  Divide the first term of the dividend (numerator) by the first term of the divisor (denominator)
                                                                \frac{2x^2}{x}=2x

5.  Multiply the divisor by that term
     2x(x+4)=2x^2+8x

6.  Subtract this from the dividend

7.  Repeat steps 4-6 until you get the remainder
     \frac{(-3x)}{x}=-3

8.  Put the remainder over the divisor to create a fraction and add it to the new polynomial
                                                                  2x-3+\frac{(-6)}{(x+4)}

Dividing polynomials using long division is very tricky. It is so easy to skip an exponent, have an algebraic error, and forget a step. This is why practicing this type of problem is so important. The only way to get better at it is to keep practicing it. Check out Symbolab’s Practice for practice problems and quizzes.

Until next time,

Leah.