Rationalising the denominator
- Multiply both the numerator and the denominator by the irrational root in the denominator (can only be done if there is only one value on the bottom)
- Multiply both the numerator and the denominator by the conjugate
Conjugate
The conjugate is where we change the sign in the middle of two terms
* We only use it in expressions with two terms, called “binomials”
Solving rational equations
- CHECK THE DOMAIN
- You can cross-fire multiply when there is an equals sign
- When adding or subtracting make them the same denominator by mutiplying the numerators by the opposite denominator and multiplying the two denominators
- If you have two fractions on either side of the equation, the two numerators will be equal
Solving irrational equation
- CHECK THE DOMAIN (there can’t be a negative under the square root)
- Isolate one of the square roots (if there are multiple)
- Solve (usually be squaring everything)
- HAVE TO CHECK SOLUTIONS since both sides were squared
Solving exponential equations
- Try to get the same base or the same exponent
- if you have the same exponent you can multiply (only multiply?) the bases and keep the exponent
- if you have the same base and an equal sign between, the exponents are equal - If the bases cannot be simplified you might have to use the natural log
Solving logarithmic equations
- Check the domain, remember the expression in the bracket has to be greater than 0
- Use log rules to simplify or use base change
Solving modular equations with an absolute value on one side of the equals and an expression or a constant on the other
Get rid of the absolute value and write two possibilities
1. Where the sign of the other expression stays the same
2. Where the sign of the other expression is reversed
Then solve and CHECK SOLUTIONS
Solving modular equations with mod on both sides of the ‘=’ sign
Remove both mods and make two possibilities
1. Where the sign of one of the expressions stays the same
2. Where the sign of one of the expressions is reversed
Then solve and CHECK SOLUTIONS
Solving modular equations using graphs
- Draw the graph of the expressions (modular or not) in the equation
- Find their points of intersections which will be the solutions to the equation
- Make the equations of the lines equal (remember: the modular graph is made of two lines, the one sloping down is multiplied by -) and solve for x(s)
- CHECK ANSWERS
Solving modular equations using their definition
- Write the positive and negative case/solution for each absolute value expression
- Draw a number line with three sections and map the possible solutions from either expression to that region
- Add up these sections “vertically”
- Form three conditional expressions for the three intervals and attach the respective sum to each
- Solve the equations and CHECK THE ANSWERS
Solving equations to the higher degree
- Try to factorise a squared exponent from the higher degree and substitute a variable into the higher degree
- Try to substitute an expression in the equation
Solving quadratic inequalities
- Reduce one side to 0
- Solve the equation for x to find critical points (zeroes)
- Sketch the graph
- Read solution from graph
