Algebra III

What are natural numbers?

Positive whole numbers

Symbol for natural numbers

N

How are natural numbers represented on a number line?

Dots on a number line

What are integers?

Positive and negative whole numbers

Symbol for integers

Z

How are integers represented on a number line?

Dots on a number line

What are rational numbers?

Fractions

Symbol of Rational numbers

Q

What are irrational numbers?

Numbers that cannot be written as a fraction e.g. surds. π

Symbol for irrational numbers

R/Q

What are real numbers?

All rational and irrational numbers

Symbol for real number

R

What does a full dot on a numberline mean?

Includes the whole number on the dot

What does a hollow dot on a numberline mean?

Does not include the whole number on the dot

When solving an equation with surds, what should you do?

Check the answer

How to solve inequalities

Treat inequality sign as an equal sign and solve for x

What happens when you divide an inequality by a negative number

Sign should point to other side

How to solve double inequalities e.g. 3 > x-2 >-5

Split them into two separate equations with the middle in both e.g. 3> x-2. x-2>-5

> means …

Above the x-axis

≥ means …

On or above the x-axis

< means ..

Below the x-axis

≤ means …

On or below the x-axis

How to cancel algebraic fractions in an inequality?

Multiply both sides by the denominator squared

What is the absolute value/modulus?

The distance of a real number away from zero on the number line

Modulus/absolute value is …

Always positive

How to cancel out modulus?

Square it and square it out

How to cancel a square root?

Square other terms, but terms in the square root don’t change

When a number is outside of a modulus bracket, what happens when you try solve it?

It also gets squared

What proof is used for inequalities? e.g. p² +1 ≥ 2p

(real)² ≥ 0

what is the discriminant? What does it do?

b²-4ac. Determines whether the graph cuts, touches, or doesn’t cut the x-axis

What does it mean when the discriminant is greater than zero? (2)

Cuts the x-axis. 2 real different roots

What does it mean when the discriminant is equal to zero? (2)

Touches the x-axis. 2 real equal roots

What does it mean when the discriminant is less than zero? (2)

Doesn’t cut the x-axis. 2 non-real (imaginary) roots