You can solve systems in two or three variables by elimination or matrices, including a line meeting a curve.
Weeks 11-12
College Algebra course units
Work in order for a complete course, or jump directly to the unit you need for an upcoming assessment.
01
Equations and inequalities
Weeks 1-2
You can solve linear, quadratic, absolute-value and compound problems and write the answers as intervals. You can also solve a formula for whichever letter you need.
Solve an absolute-value equation by splitting it into two cases.
Write -3 < x and x <= 7 as the interval (-3, 7].
Solve a quadratic inequality by testing signs on each side of the roots.
Worked example
Solve the absolute value of (2x - 5) = 9
Case 1: 2x - 5 = 9, so x = 7
Case 2: 2x - 5 = -9, so x = -2
Answerx = 7 or x = -2
Most common mistakeSolving the positive case only: reporting x = 7 alone and losing x = -2, which also sits exactly 9 units away.
You can use function notation, find domain and range, and build new functions out of old ones. You can also evaluate a piecewise rule on either side of its break.
Find the domain of a square-root function.
Evaluate a piecewise function just below and just above the break point.
Compute (f - g)(2) from two given formulas.
Worked example
Find the domain of f(x) = the square root of (x - 4)
What is under the root cannot be negative: x - 4 is 0 or more
x is 4 or more
AnswerThe interval [4, infinity)
Most common mistakeRequiring the inside to be strictly positive: writing (4, infinity) and throwing out x = 4, even though f(4) = 0 is a perfectly good value.
You can find every root of a polynomial, including repeated ones, and sketch the graph from them. You can also use a polynomial to model a volume or a profit.
List the possible rational roots with the p over q rule.
Explain why a doubled root makes the graph touch the axis and bounce off.
Build a profit polynomial as revenue minus cost.
Worked example
Factor x cubed - 4x squared + 4x and list the roots
Take out the common x: x(x squared - 4x + 4)
x(x - 2)(x - 2)
AnswerRoots x = 0, and x = 2 counted twice
Most common mistakeDividing both sides by x to clear it: that throws away the root x = 0 and leaves only x = 2, so the graph's crossing at the origin disappears.
You can work with formulas that have variables in the denominator, including work-rate and mixture problems. You can also describe asymptotes and end behaviour.
Solve a work problem where two pipes fill one tank together.
Find a slant asymptote by long division.
Reject any answer that makes a denominator zero.
Worked example
Pipe A fills a tank in 6 hours, pipe B in 3 hours. How long together?
Rates add: 1/6 + 1/3 = 1/6 + 2/6
3/6 = 1/2 tank per hour
Answer2 hours
Most common mistakeAveraging the two times: reporting (6 + 3)/2 = 4.5 hours, which is slower than pipe B managed on its own.
You can model interest, population and cooling with exponentials, then undo them with logs. You can also compare yearly, quarterly and continuous compounding.
Use A = P(1 + r/n) to the power nt for quarterly compounding.
Solve for the time it takes an amount to double.
Convert between a continuous rate and a yearly percentage.
Worked example
2,000 invested at 5% compounded quarterly. Find the value after 8 years.
A = 2000(1 + 0.05/4) to the power (4 x 8)
A = 2000 x 1.0125 to the power 32 = 2000 x 1.4881
AnswerAbout 2,976.20
Most common mistakeForgetting to divide the rate by 4: using 1.05 to the power 32 and reporting about 9,682, more than three times the true amount.
You can solve systems in two or three variables by elimination or matrices, including a line meeting a curve. You can also shade the overlap for a system of inequalities.
Solve a three-variable system by eliminating the same variable twice.
Solve a line-and-circle system by substitution.
Shade the region where two inequalities are both true.
Worked example
Solve y = x + 1 together with x squared + y squared = 25
Substitute: x squared + (x + 1) squared = 25
2x squared + 2x + 1 = 25, so x squared + x - 12 = 0
Answer(x + 4)(x - 3) = 0, giving the points (3, 4) and (-4, -3)
Most common mistakeExpanding (x + 1) squared as x squared + 1: the equation collapses to 2x squared = 24 and returns points that are not on the circle at all.
Before timing yourself, check whether you can explain Equations and inequalities from a blank page. Then connect it to Functions. If either explanation depends on copying a formula, review the unit first and complete two untimed examples.
Use tools to verify, not to choose the method for you
The Algebra calculator can test calculations and representations used in College Algebra. Make the setup yourself, predict the sign or scale, and compare the tool result with that prediction. Use the formula library to check conditions as well as notation.
Know when to move to the full test
Move from College Algebra practice to the complete course test after you can correct a missed problem without reopening the worked answer. Record the earliest wrong decision—not only the final score—so the next study session has a precise target.
Before and after the syllabus
Learn the ideas, then practise them
The unit list tells you what is covered. These two pages are where the method is explained and where you find out whether it stuck.
Start with Equations and inequalities if you are following the full sequence. If that unit feels automatic, open the College Algebra practice page, choose mixed review, and let the first errors identify the earliest prerequisite to revisit.
How do I know I am ready for the course test?
You are ready when you can choose a method without a hint, show the governing steps, and explain why the result is reasonable. Use the complete College Algebra test only after you can correct practice errors from a blank page.
Which calculator supports this course?
The Algebra calculator supports the calculations and representations used in this course. Use it to test or visualize a result after making your own setup.
