Lecture: Midterm Review & Exam
Lecture script · about 15–20 minutes spoken · Week 8
Opening
Think about a nurse who checks a dosage calculation before giving it, or a small-business owner deciding whether a pricing question needs one equation or two. Before either of them does any arithmetic, they read the problem and decide what kind of problem it is. That first decision is the subject of this lecture. It's the habit that makes a mixed test feel calm instead of frantic.
Here's where we are in this course. In Week 1 you learned the order of operations and the exponent rules. In Week 2 you solved equations and inequalities. In Week 3 you met functions, in Week 4 you read lines, and in Week 5 you solved systems. Week 6 multiplied and factored polynomials, and Week 7 solved quadratic equations. The midterm covers all seven of those weeks. After it, in Week 9, the parabola you solved in Week 7 becomes a graph.
So this lecture does two jobs. First, I'll walk through six toolboxes, one per core move, with one fresh problem each. Then I'll teach the one new habit of the week: reading a problem's cue and naming its move before you start. Every problem ends with a check, because a check is what turns a guess into a result.
1. Simplify: follow the order and the exponent rules
Picture a receipt with no total at the bottom. Nobody is asking you to find a mystery number. You're only asking what the receipt adds up to. That's what a problem with no equals sign wants: simplify it, or evaluate it.
Here's the first one. Negative 2 cubed, plus negative 3 in parentheses, squared. The order of operations says exponents come before that leading minus sign. So negative 2 cubed means the opposite of 2 cubed, which is negative 8. The second term is different. The parentheses put the minus sign inside the base, so negative 3 times negative 3 is positive 9. Negative 8 plus 9 is 1.
The second one is 2 x squared y, all raised to the third power. The power of a product says every factor gets raised to that power. Two cubed is 8. Then x squared, raised to the third, is x to the 6th, because a power of a power multiplies the exponents: 2 times 3 is 6. And y cubed stays y cubed. Put together, you get 8 x to the 6th, y cubed.
Here's how to check it. Put in x equals 2 and y equals 1. The original becomes 8 cubed, which is 512. The answer becomes 8 times 64, which is also 512. A wrong exponent of 5 would give 8 times 32, which is 256, and that's the slip the check catches.
Pause and try this. Someone simplifies negative 3, in parentheses, squared, and writes negative 9. What did they do wrong?
Give it a moment before I go on.
Here's the answer. They dropped the parentheses. With the parentheses, the base is negative 3, and negative 3 times negative 3 is positive 9. Negative 9 belongs to negative 3 squared with no parentheses, where the minus sign waits its turn.
2. Solve one-variable statements: equation, inequality, absolute value
Let's go to two bike shops on the same street. Shop A lets you ride the first hour free and then charges 4 dollars for each hour after that. Shop B charges a 6-dollar fee plus 2 dollars an hour. An equals sign between their prices asks for the one number of hours where they match.
So 4 times the quantity x minus 1 equals 2x plus 6. Distribute to get 4x minus 4 equals 2x plus 6. Subtract 2x from both sides, add 4, and you have 2x equals 10, so x is 5. Check it. Shop A costs 4 times 4, which is 16 dollars. Shop B costs 10 plus 6, which is 16 dollars. At 5 hours the two shops cost the same.
An inequality sign asks for a whole set of numbers. Take negative 3x plus 2 is less than or equal to 11. Subtract 2 to get negative 3x is less than or equal to 9. Now divide by negative 3. This is the moment that matters: dividing by a negative number flips the sign. So x is greater than or equal to negative 3. In interval notation, that's a bracket at negative 3 and a parenthesis at infinity.
Absolute-value bars call for the split rule. The absolute value of 2x minus 3 equals 7 splits into two equations: 2x minus 3 equals 7, or 2x minus 3 equals negative 7. The first gives x equals 5, and the second gives x equals negative 2. Each one gets its own check, and both land on 7.
Pause and try this. If you forget to flip the sign in that inequality, you get x is less than or equal to negative 3. Test x equals negative 4 in the original. Does it work?
Try the arithmetic in your head.
Here's the answer. It doesn't work. Negative 3 times negative 4 is 12, and 12 plus 2 is 14. And 14 is not less than or equal to 11, so the unflipped set includes a number that fails, and that's how you know the flip was needed.
3. Work with functions: evaluate, find a domain, compose
A vending machine is a good picture of a function. You press one button, and exactly one snack comes out. Function notation writes the output as f of x. It's one value, never f times x.
Let f of x be x squared minus 2x, and evaluate it at negative 3. Wrap the input in parentheses every time. Negative 3, squared, is 9. Negative 2 times negative 3 is positive 6. So 9 plus 6 gives f of negative 3 equals 15.
Next, a domain is the set of every input a function can accept and still give a real-number output. A square root needs what's inside it to be zero or more. So the square root of x minus 4 needs x minus 4 to be at least 0, which means x is at least 4. The domain is the interval from 4 to infinity, with a bracket at 4, because the square root of 0 is 0, a perfectly good output.
Last, composition works inside first. Take p of x equals x squared and q of x equals x plus 1. For p of q of 2, you do q first: 2 plus 1 is 3, and 3 squared is 9. For q of p of 2, you do p first: 2 squared is 4, and 4 plus 1 is 5. Same two functions, different order, different result.
Pause and try this. Someone writes the domain of the square root of x minus 4 with a parenthesis at 4 instead of a bracket. Is 4 in the domain or not?
Think about what happens right at 4.
Here's the answer. Four is in the domain. At x equals 4, the inside is 0, and the square root of 0 is 0, a real number. The rule is zero or more, not strictly more, so 4 gets a bracket.
4. Read lines and solve systems of two equations
Picture a cool morning that warms up steadily. One hour before noon, the thermometer reads 2 degrees Celsius. Three hours after noon, it reads 10 degrees. Two points and a steady rate are the cue for a line.
Let x be the hours after noon, so one hour before noon is x equals negative 1. The slope, the steepness of a line, is rise over run with the y-values on top. The rise is 10 minus 2, which is 8. The run is 3 minus negative 1, which is 4. Eight divided by 4 is 2, so the temperature climbs 2 degrees per hour. Building the line from the first reading and the slope gives y equals 2x plus 4. That 4 is the y-intercept, and it says the thermometer read 4 degrees at noon. Check the reading three hours after noon: 2 times 3 plus 4 is 10.
Now two equations at once. A 7-hour workday is split between two tasks, and one task takes 1 hour longer than the other. That gives x plus y equals 7 and x minus y equals 1. Add the two equations, and the y's cancel, so 2x equals 8 and x is 4. Then y is 3. A solution of a system has to work in both equations, so check both: 4 plus 3 is 7, and 4 minus 3 is 1.
Pause and try this. Someone finds the run as 3 minus 1 instead of 3 minus negative 1. What slope do they get, and why is it wrong?
Work out their version first.
Here's the answer. They get 8 divided by 2, which is 4. The second point's x-value is negative 1, so the run is 3 minus negative 1, and subtracting a negative adds. The run is 4, so the slope is 2.
5. Multiply and factor polynomials
Imagine a square rug that's 2x plus 3 feet on each side. Its area is that side times itself, so you need 2x plus 3, squared. A square of a sum follows the special product: the first term squared, plus twice the product of the two terms, plus the last term squared.
So 2x squared is 4x squared. Twice 2x times 3 is 12x. And 3 squared is 9. The rug covers 4x squared plus 12x plus 9 square feet. Check at x equals 1. The side is 5, and 5 squared is 25. The polynomial gives 4 plus 12 plus 9, which is also 25.
Factoring runs that work backward. Take x squared minus 6x plus 5, the parabola you've met all through Weeks 6 and 7. You need two numbers whose product is 5 and whose sum is negative 6. Negative 1 and negative 5 do it. So the factors are x minus 1 and x minus 5. Multiply them back out and you get x squared minus 6x plus 5 again, which is your check.
Pause and try this. Someone writes 2x plus 3, squared, as 4x squared plus 9. Which piece is missing, and what does the check at x equals 1 show?
Picture the two numbers side by side.
Here's the answer. The middle term, 12x, is missing. At x equals 1, the true area is 25, but their version gives 4 plus 9, which is 13. A square of a sum always has a middle.
6. Solve a quadratic equation four ways, discriminant first
Here's a puzzle. A number, squared, minus 6 times the number, plus 5, equals 0. A squared variable in an equation is the cue for a quadratic equation. Before you solve it, read the discriminant, the quantity b squared minus 4ac. Here a is 1, b is negative 6, and c is 5. Negative 6, squared, is 36, and 4 times 1 times 5 is 20, so the discriminant is 16. It's positive, so you expect two real solutions.
Now three paths. By factoring, x minus 1 times x minus 5 equals 0, so x is 1 or 5. By completing the square, you move the 5 across, take half of negative 6, which is negative 3, and square it to get 9. Adding 9 to both sides turns the left side into x minus 3, squared, and the right side into 4. The square root property finishes it: x minus 3 is plus or minus 2, so x is 5 or 1. By the formula, you get 6 plus or minus 4, all over 2, which is 5 or 1. Three paths, one answer. And the check: 1 minus 6 plus 5 is 0, and 25 minus 30 plus 5 is 0.
The discriminant can also tell you to stop. For x squared plus 2x plus 5 equals 0, it's 4 minus 20, which is negative 16, so there's no real solution. For 2x squared minus 4x plus 2 equals 0, it's 16 minus 16, which is 0, so there's one repeated solution, x equals 1.
Pause and try this. Using the formula on the puzzle, someone gets 6 plus or minus 4 on top and then divides only the 4 by 2. What do they write, and what's the fix?
Hold the fraction bar in your mind.
Here's the answer. They write 6 plus or minus 2, which gives 8 and 4. The fraction bar covers the whole top, so 10 divided by 2 is 5, and 2 divided by 2 is 1.
7. Choose a method: name the core move before you start
Picture a mechanic who listens to an engine for a few seconds before opening the hood. That pause decides which tool comes out first. Choosing a method is the habit of reading a problem's cue, naming its core move, and picking a procedure before starting, so the first step is chosen on purpose rather than by guessing.
The cue is how the problem is written. No equals sign means simplify. An equals sign with x to the first power means solve. An inequality sign means solve and watch for a flip. Absolute-value bars mean split. A function name around a number means evaluate. Two points mean slope. Two equations in two unknowns mean a system. A squared binomial means multiply, and the word factor means run it backward. A squared variable in an equation means a quadratic.
For a quadratic, the form picks the method. X squared equals 49 is a square equal to a number, so the square root property gives plus or minus 7. Three x squared minus 12x equals 0 has no constant term, so you factor out 3x and get 0 or 4. Never divide by x there, or you lose the 0. X squared minus 4x plus 1 equals 0 has a discriminant of 12, which isn't a perfect square, so the formula gives 2 plus or minus the square root of 3.
One more, from Week 7. The launched object lands when negative 16 t squared plus 64t plus 80 equals 0. Every coefficient divides by negative 16, so you do that first and get t squared minus 4t minus 5 equals 0. That factors into t minus 5 times t plus 1. So t is 5 or negative 1, and negative 1 is before the launch, so you reject it. The object lands at 5 seconds.
Pause and try this. Name the core move for each of these, without solving: the absolute value of 2x minus 3 equals 7, and x plus y equals 7 with x minus y equals 1.
Say each move in one word if you can.
Here's the answer. The first is a split, because absolute-value bars equal a positive number. The second is a system, because there are two equations in two unknowns.
Closing
Let me pull it together. To simplify, follow the order of operations and keep a negated power apart from a negative base. To solve, do the same thing to both sides, flip on a negative divisor, and split an absolute value. With a function, wrap the input in parentheses, check the edge of a domain, and work inside first. With a line, put the y-values on top, and check a system's pair in both equations. With a polynomial, keep the middle term, and expand to check a factorization. With a quadratic, read the discriminant first and let the form pick the method. And before any of it, read the cue and name the move.
Later in this module, the slide deck, the Readings & Resources page, the practice set, and the discussion give you more chances to sort problems by their moves, and then comes the midterm. Next week, in Week 9, Quadratic Functions & Their Graphs, the parabola you solved in this lecture becomes a picture you can read.
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