# 1.2. Precalculus: Algebra

### Learning Objectives

You should be able to

• solve equations by rearrangement or using factorised forms
• expand brackets and simplify algebraic expressions
• solve inequalities and write solution in interval notation
• solve systems of simultaneous inequalities
• plot regions of the plane which are solutions to simultaneous linear inequalities

### Get Started: What to do next

If you can pass the diagnostic test, then we believe you are ready to move on from this topic.

Our recommendation is to:
1. Test your ability by trying the diagnostic quiz. You can reattempt it as many times as you want and can leave it part-way through.
2. Identify which topics need more work. Make a note of any areas that you are unable to complete in the diagnostic quiz, or areas that you don't feel comfortable with.
3. Watch the tutorials for these topics, you can find these below. Then try some practice questions from the mini quiz for that specific topic. If you have questions, please ask on the forum.
4. Re-try the diagnostic quiz. And repeat the above steps as necessary until you can pass the diagnostic test. A pass grade is 80%.

Optionally, also see if you can apply these skills to an economic application or look at the additional resources and advanced quizzes at the bottom of the page,

### Tutorials: How to guides

#### 1.2.1. Solving equations

See how to solve equations through the process of rearranging and undoing what has been done to the variable, how to use factorised forms and some simultaneous equations.

#### 1.2.2. Expanding brackets

$$c(x+y)=cx+cy$$,      $$(a+b)(x+y) = ax+ay+bx+by$$,
$$(x+y)^2 = x^2+2xy+y^2$$,      $$x^2-y^2= (x-y)(x+y)$$,
$$(x+y)^3 = x^3+3x^2y+3xy^2+y^3$$

#### 1.2.3. Inequalities

How to solve & manipulate inequalities and how to express solutions in interval notation.
$$a<b \implies a+c<b+c, a-c<b-c, b>a$$
$$c>0, a<b \implies ac<bc, a/c < b/c$$
$$c<0, a<b \implies ac<bc, a/c < b/c$$
$$a<x\leq b \iff x \text{ is in } (a,b]$$

#### 1.2.4. Simultaneous Inequalities

How to solve systems of simultaneous inequalities and examples of where they might arise.

### Further practice & resources

You can find practice quizzes on each topic in the links above the tutorial videos, or you can take the diagnostic quiz as many times as you like to. The additional resources gives links to external sources of information and some of these may also have further exercises to try.