Algebra and Expressions

Algebra generally occurs in the form of:

• words, or,
• already expressed calculations.

In algebra the letters that you see are representatives of an unknown number. They're called variables and you can also substitute in numbers there.

Algebra is kind of like a recipe (without the ingredient list), it shows you the steps (the method) but the ingredient list is left for you to figure out.

There are ways that algebra is expressed and most often, algebra is shown in its simplest form (for ease).

Here are some general rules:

• Addition – same variable, just add together. E.g. 2x + 5x + 2 = 7x + 2, x² + 4x + x = x² + 5x
• Subtraction – same variable, just subtract. E.g. 5x – 2x + 1 = 3x + 1
• Multiplication – combine together. E.g 5 ⋅ x ⋅ 6x ⋅ 2y = 60x²y (Dot operator [⋅] denotes multiplication as using “x” may be confused with the actual “x” variable).
• Division – 2x / 4x = 1/2, 2x/5x = 2/5, 2x/5y (this can’t be simplified further).

You can even substitute in numbers (ingredients) to see if the formula/expression works (recipe).

E.g. Let’s substitute “1” for x.

2x + 5x + 2 = 2 + 5 + 2 = 9

7x + 2 = 7 + 2 = 9

Working with equations

The key to solving equations is isolate out the variable. It pretty much involves using the opposite sign when moving things to the other side. Let’s demonstrate with the following examples (see video).

• 2y + 4 = 20
• y + 20 = 50
• y – 20 = 5
• 4y / 10 = 4/5

Working with inequalities

This is a similar process to working with equations however, when multiplying or dividing both sides of an inequality by a negative number, the direction of the inequality must change.

The solution of an inequality will be a range of values for the variable, rather than just one value. Let’s demonstrate with the following examples (see video).

• 2y + 4 < 20
• y + 20 > 50
• y – 20 < 5
• 4y / 10 > 4/5

Example Question/s

Watch video for explanation of the following question/s:

Which inequation shows the following statement? x is greater than 7 but less than or equal to 10

A 10 > x > 7 B 10 < x < 7 C 10 < x > 7 D 10 < x > 7 E 10 < x > 7

If V= [(x-y)/4] then x equals

A Y + 4V B Y - V C Y/4 + V D Y + V/4 E 4(Y + V)

Solve the inequation for y

9(y + [2/3]) < 7

A Y > -1/9 B Y < -1/9 C Y > 1/9 D Y< 1/9 E None of these

Key Rules to remember

• Addition in expression? If same variable, just add together.
• Subtraction in expression? If same variable, just subtract.
• Multiplication in expression? Combine together.
• Division in expression? Cancel out like terms and reduce by common factors.
• The key to solving equations is isolate out the variable and use the opposite sign when moving things to the other side.
• For inequalities, use a similar process to working with equations however, when multiplying or dividing both sides of an inequality by a negative number, the direction of the inequality must change.

Practice time!

Now, it's your turn to practice.

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