MrE's Algebra Blog

Algebra: Chapter 4, Lesson 2, page 175 and Lesson 3, page 180.

The Addition Property of Inequalities

Don’t forget, the equations:

• if `a < b`, then `a + c < b + c`
• if `a > b`, then `a + c > b + c`

and similar statements are true for ≤ and ≥

For inequality of one step, follow the EXACT same steps as equalities. The only things we have to remember when graphing on a number line:

• For the symbols ≤ and ≥, the circle must be CLOSED because we INCLUDE the data point
• For the symbols < and >, the circle must be OPEN because we get as close as possible to the data point but it is NOT INCLUDED!

Here is a link from PURPLEMATH.com with some more examples!

The Multiplication Property of Inequalities

The property states if `c` is POSITIVE

• if `a < b`, then `ac < bc` and
• if `a > b`, then `ac > bc`

Where `c` is NEGATIVE

• if `a < b`, then `ac > bc` and
• if `a > b`, then `ac < bc`

Following the EXACT same steps as equalities, we have learned to solve 1 step equations with inequalities. The ONLY difference is when multiplying or dividing by a NEGATIVE number, we must REVERSE the sign of the inequality for the final solution!! If we divide or multiply by a positive number, we leave the inequality sign alone.

Here are some examples from purplemath.com that have to do with inequalities with products and divisions.

Two of tonight’s homework problems for Chapter 4-2 are solved by MrE are here! Just click it! Two more are here for Chapter 4-3 as well!

======================================================================

Algebra 1a: Chapter 2, Lesson 7, page 89.

Using the Distributive Property

The distributive property of multiplication over addition and subtraction is:

`a(b+c)=ab+ac` and `a(b−c)=ab−ac`

You can also combine LIKE TERMS ONLY IF THE VARIABLE AND THE EXPONENT OF THE VARIABLE ARE EXACTLY THE SAME!

Examples, (remember to use the wacky arrows!):

`9(x – 5) = 9x – 9(5) = 9x – 45`

`-4(x – 2y + 3z) = -4x +(-4)(-2y) + (-4)(3z) = -4x -4(-2y) -4(3z)`

`= -4x + 8y – 12z`

(if it comes up, don’t forget the change-change or bling-bling, double negative thingy …)

Here is a link from purplemath.com about the distributive property with more examples.

Two of tonight’s homework problems solved by MrE are here! Just click it!

Algebra: Chapter 3, Benchmark #3

No homework!

======================================================================

Algebra 1a: Chapter 2, Lesson 7, page 89.

Using the Distributive Property

The distributive property of multiplication over addition and subtraction is:

`a(b+c)=ab+ac` and `a(b−c)=ab−ac`

You can also combine LIKE TERMS ONLY IF THE VARIABLE AND THE EXPONENT OF THE VARIABLE ARE EXACTLY THE SAME!

Examples, (remember to use the wacky arrows!):

`9(x – 5) = 9x – 9(5) = 9x – 45`

`-4(x – 2y + 3z) = -4x +(-4)(-2y) + (-4)(3z) = -4x -4(-2y) -4(3z)`

`= -4x + 8y – 12z`

(if it comes up, don’t forget the change-change or bling-bling, double negative thingy …)

Here is a link from purplemath.com about the distributive property with more examples.

Two of tonight’s homework problems solved by MrE are here! Just click it!

Algebra: Chapter 3 Review

Make sure that your notes are up-to-date.

Remember too, the steps to SOLVING 2-STEP EQUATIONS:

1. Multiply both sides to clear fractions or decimals, if necessary.
2. Collect like terms on each side, if necessary.
3. Use the addition property to move the variable to one side and all other terms to the other side of the equation.
4. Collect like terms again, if necessary
5. Add or subtract to isolate the variable and finally
6. Use the multiplication or division or reciprocal properties to solve for the variable.

======================================================================

Algebra 1a: Chapter 2,  Lesson 6, page 81.

Division of Rational Numbers

Today is easy because the rules for multiplication and division are simple.

• When multiplying 2 numbers AND if the SIGNS are the same, the product is ALWAYS positive.
• If the signs are different, them the product is ALWAYS negative. This is pretty straightforward.

Division follows the same rules as multiplication.

2 rational numbers whose product is 1 are called multiplicative inverses or reciprocals of each other. Just flip the rational expression over and keep the same sign. For example, the reciprocal of `2/3` or `m/n` is `3/2 ` and `n/m` respectively.

Remember too, to divide rational numbers, sometimes its easier to express them as improper fractions, then convert the 2nd term to its reciprocal and change the `/` to a `⋅`.

Two of tonight’s homework problems solved by MrE are here! Just click it!

Purplemath.com has these tutorials about multiplying and dividing rational numbers, check it out!

Algebra: Chapter 3 Review

Skills Practice 8 and 9, pages 16 and 17.

Make sure that your notes are up-to-date.

Bring your problems worked out and your questions for the Chapter 3 review. These type of questions will appear on the test. Make sure that your notes are up-to-date. Practice makes perfect

Remember how to do the whickity-whack divide thingy.

Remember too, the steps to SOLVING 2-STEP EQUATIONS:

1. Multiply both sides to clear fractions or decimals, if necessary.
2. Collect like terms on each side, if necessary.
3. Use the addition property to move the variable to one side and all other terms to the other side of the equation.
4. Collect like terms again, if necessary
5. Add or subtract to isolate the variable and finally
6. Use the multiplication or division or reciprocal properties to solve for the variable.

======================================================================

Algebra 1a: Chapter 2, Lessons 5 and Lesson 6, pages 77 and 81.

Multiplication and Division of Rational Numbers

Today is easy because the rules for multiplication and division are simple.

• When multiplying 2 numbers AND if the SIGNS are the same, the product is ALWAYS positive.
• If the signs are different, them the product is ALWAYS negative. This is pretty straightforward.

Division follows the same rules as multiplication.

2 rational numbers whose product is 1 are called multiplicative inverses or reciprocals of each other. Just flip the rational expression over and keep the same sign. For example, the reciprocal of `2/3` or `m/n` is `3/2 ` and `n/m` respectively.

Remember too, to divide rational numbers, sometimes its easier to express them as improper fractions, then convert the 2nd term to its reciprocal and change the `/` to a `⋅`.

Two of tonight’s homework problems solved by MrE are here! Just click it!

Purplemath.com has these tutorials about multiplying and dividing rational numbers, check it out!

Algebra: Chapter 3, Lesson 10, page 152 and Chapter 3, Lesson 11, page 158.

More Expressions and Equations

Remember, the sum of an integer and the next integer can be represented by `x` and `(x+1)` or `x+(x+1)` or `2x+1`.

The sum of consecutive (comes right after each other) odd OR even integers can be expressed as `x` and `(x+2)` or `x+(x+2)` or `2x +2`.

If you get confused, just make a little table, like `3`, `4`, `5`, `6`, `7` and `8` and see where the variable `n ` would line up if the numbers were hidden.

Here is a link that shows a few examples too.

Two of tonight’s homework problems solved by MrE are here! Just click it!

Using Percent
The ratio of numbers to 100 is called percent. Percent means “per one hundred”. We use whickity-whack divide, the method that Ms. Phillips taught us in 7th grade. Here is my podcast that describes the process!

Two of tonight’s homework problems solved by MrE are here! Just click it!

======================================================================

Algebra 1a: Chapter 2, Lessons 5 and Lesson 6, pages 77 and 81.

Multiplication and Division of Rational Numbers

Today is easy because the rules for multiplication and division are simple.

• When multiplying 2 numbers AND if the SIGNS are the same, the product is ALWAYS positive.
• If the signs are different, them the product is ALWAYS negative. This is pretty straightforward.

Division follows the same rules as multiplication.

2 rational numbers whose product is 1 are called multiplicative inverses or reciprocals of each other. Just flip the rational expression over and keep the same sign. For example, the reciprocal of `2/3` or `m/n` is `3/2 ` and `n/m` respectively.

Remember too, to divide rational numbers, sometimes its easier to express them as improper fractions, then convert the 2nd term to its reciprocal and change the `/` to a `⋅`.

Two of tonight’s homework problems solved by MrE are here! Just click it!

Purplemath.com has these tutorials about multiplying and dividing rational numbers, check it out!

Algebra: Chapter 3, Lesson 9, page 148.

Proportions

A ratio of 2 quantities is a comparison, often expressed as a fraction. An equation that states that 2 ratios are equal is called a proportion. I prefer to just criss-cross, or cross multiply proportional problems, but the book’s way is OK too.

For example,

`x/63=2/9`, I solve by criss-crossing. That becomes

`x*9=2*63` or `9x=63*2` or `9x=126`

and dividing both sides by 9 to clear the x, gives us `x=14`.

Here is a link from purplemath.com that has some more examples.

Two of tonight’s homework problems solved by MrE are here! Just click it!

======================================================================

Algebra 1a: Chapter 2, Lessons 5 and Lesson 6, pages 77 and 81

Multiplication and Division of Rational Numbers

Today is easy because the rules for multiplication and division are simple.

• When multiplying 2 numbers AND if the SIGNS are the same, the product is ALWAYS positive.
• If the signs are different, them the product is ALWAYS negative. This is pretty straightforward.

Division follows the same rules as multiplication.

2 rational numbers whose product is 1 are called multiplicative inverses or reciprocals of each other. Just flip the rational expression over and keep the same sign. For example, the reciprocal of `2/3` or `m/n` is `3/2 ` and `n/m` respectively.

Remember too, to divide rational numbers, sometimes its easier to express them as improper fractions, then convert the 2nd term to its reciprocal and change the `/` to a `⋅`.

Two of tonight’s homework problems solved by MrE are here! Just click it!

Purplemath.com has these tutorials about multiplying and dividing rational numbers, check it out!

Algebra: Chapter 3, Lesson 8, page 145.

Solving Equations Involving Absolute Value

The absolute value of a number, `|x|` is its distance from zero on the number line. Remember that the absolute value is always positive. Treat absolute value equations just like those without absolute value and then solve them as normal. At the VERY END, put the absolute value symbols back in and see if the answer has ANOTHER solution.

Remember too, that we cannot have an absolute value be NEGATIVE. In these cases, there is NO SOLUTION. For example:

`|x| + 2 = 12`

`|x| + 2 + (-2) = 12 + (-2)`, we subtract 2 from both sides to isolate the variable, x

`|x| = 10`, we simplify the right side and finally,

`x=10` or `x=-10` are the solutions

This is a good purplemath.com link with examples.

Two of tonight’s homework problems solved by MrE are here! Just click it!

======================================================================

Algebra 1a: Chapter 2, Lessons 5 and Lesson 6, pages 77 and 81.

Multiplication and Division of Rational Numbers

Today is easy because the rules for multiplication and division are simple.

• When multiplying 2 numbers AND if the SIGNS are the same, the product is ALWAYS positive.
• If the signs are different, them the product is ALWAYS negative. This is pretty straightforward.

Division follows the same rules as multiplication.

2 rational numbers whose product is 1 are called multiplicative inverses or reciprocals of each other. Just flip the rational expression over and keep the same sign. For example, the reciprocal of `2/3` or `m/n` is `3/2 ` and `n/m` respectively.

Remember too, to divide rational numbers, sometimes its easier to express them as improper fractions, then convert the 2nd term to its reciprocal and change the `/` to a `⋅`.

Two of tonight’s homework problems solved by MrE are here! Just click it!

Purplemath.com has these tutorials about multiplying and dividing rational numbers, check it out!

Algebra: Chapter 3-7, page 142.

Formulas

Formulas that use more than 1 letter are often called literal equation. Use the formula and solve for the desired variable by treating all other variables as if they are constants or coefficients. Constants are separated by `+` and `−` symbols whereas coefficients are like `2x`, the `2` is a coefficient.

Just solve these like any other equation using the strategies from yesterday

1. Multiply both sides to clear fractions or decimals, if necessary.
2. Collect like terms on each side, if necessary.
3. Use the addition property to move the variable to one side and all other terms to the other side of the equation.
4. Collect like terms again, if necessary.
5. Use the multiplication property to solve for the variable.

Two of tonight’s homework problems solved by MrE are here! Just click it!

======================================================================

Algebra 1a: Chapter 2, Lesson 4, page 71.

Subtraction of Rational Numbers

Today, we are working on subtraction of rational numbers. Remember, just do the inverse of subtraction and ADD the inverse (or opposite) of the number to the other number.

Examples:

`2-6 = 2 + (-6) = -4`

`-5 – 7` is really `-5 + (-7)`, remembering that the + sign is invisible. So this becomes just `-5 + (-7)` and remembering the adding rule of 2 negatives, the answer becomes `-12`.

`-4 – (-5)` is really `-4 + 5` because 2 negative are a positive (bling-bling or make change-change from Ms. Phillips and Ms. Craig) and the answer is `+5 – 4` and the final answer is `+1`!

Two of tonight’s homework problems solved by MrE are here! Just click it!

Purplemath.com has these tutorials for the addition and subtraction of rational numbers too!

Algebra: Chapter 3-5 and 3-6, pages 136 and 139.

More on Solving Equations (3-5)

Solve equations by first getting all variables on the same side of the equation. Get rid of the paranthesis too (distributive property if required). Go slow and ONCE YOU HAVE YOUR ANSWER, PLUG IT BACK IN TO SEE IF IT WORKS, IF NOT, CHECK YOUR MATH FOR SIMPLE MATH OR SIGN ERRORS!

Click here for some more examples. Don’t forget to try out Nutshellmathplus.com too!

Clearing an Equation of Fractions or Decimals (3-6)

In equations containing fractions, you can use the multiplication property to make the equation easier to solve. To clear the equation of fractions, multiply both sides of the equation by the least common denominator (LCD) of all the fractions in the equation. If you wish to clear the decimals in an equation, multiply both sides by the appropriate power of 10 OR move the decimal places to the left (or right as necessary) for ALL terms an equal amount (e.g., make sure you move them ALL 2 places to the left – obviously, this is the same as multiplying by 100)

Remember the steps to SOLVING EQUATIONS:

1. Multiply both sides to clear fractions or decimals, if necessary.
2. Collect like terms on each side, if necessary.
3. Use the addition property to move the variable to one side and all other terms to the other side of the equation.
4. Collect like terms again, if necessary
5. Add or subtract to isolate the variable and finally
6. Use the multiplication or division or reciprocal properties to solve for the variable.

Two of tonight’s homework problems solved by MrE are here! Just click it and here!

======================================================================

Algebra 1a: Chapter 2, Lesson 3, page 63.

Addition of Rational Numbers.

Adding 2 positive or negative numbers

• Add the absolute values. The sum has the same sign as the 2 numbers.

Adding a positive and a negative number

• Subtract the absolute values. The sum has the sign of the number with the bigger sign!

I have a podcast on iTunes that talks about this and tomorrow’s lesson. Just search on iTunes “MrE Algebra” and you’ll find it. Click here and you can check it out here on your desktop!

Two of tonight’s homework problems solved by MrE are here! Just click it!

If you need a refresher, please read Chapter 2, lessons 1 and 2 or click here!

Purplemath.com has these tutorials for the addition and subtraction of rational numbers too!

Algebra: Chapter 3-3 and 3-4, pages 125 and 130

Using the Properties Together

You can solve these equations in a variety of ways. Remember the objective is to ISOLATE the variable on one side. You can add, subtract, multiply or divide the same thing on both sides of the equation.

Follow these 4 steps:

1. you should solve equations with parenthesis using the distributive property
2. if there are like terms on one side of the equation, collect those first.
3. add and subtract the constants
4. finally, multiply or divide by the coefficient (next to the variable) to isolate the variable

Practice makes perfect and be sure to SHOW ALL THE STEPS! Purplemath has some examples here about these multi-step equations.

Don’t forget to use the reciprocal, its sometimes the same as dividing but faster.
DO NOT TAKE ANY SHORTCUTS, SHOW ALL THE WORK IN ALL ITS GORY DETAIL, THIS WILL REALLY SAVE YOU IN THE LONG RUN BY CUTTING DOWN ON SILLY MISTAKES!!

Expressions and Equations

The phrase the quantity suggests a grouping of terms will follow. The words sum of, difference of , product of and quotient of also suggests a grouping of terms (USING PARENTHESES) to follow.

The quantity “3 less than a number” is written `n−3`. The text “4 times the quantity 3 greater than a number” is translated to `4(n+3)`.

Finally, here are some problem-solving guidelines:

Phase 1: UNDERSTAND THE PROBLEM

• What am I trying to find?
• What data am I given?
• Have I ever solved a similar problem?

Phase 2: Develop and carry out a PLAN

• What strategies might I use to solve the problem?
• How can I correctly carry out the strategies I select?

Phase 3: Find the ANSWER and CHECK

Two of tonight’s homework problems solved by MrE are here! Just click it!

======================================================================

Algebra 1a: Chapter 2, Lesson 2, page 59.

Rational Numbers

“Any number that can be expressed as the ratio of two integers, `a/b`, is called a rational number.

There is a point on the number line for every rational number. The number is called the coordinate of the point and the point is the graph of the number. When we draw a point for a number on a number line, we say that we have graphed the number.“