Linear Equations, Models, and Inequalities

This is Unit Three of College Algebra: linear equations, models, and inequalities. The capstone states the unit's principle at the end, and it is short enough to carry from the beginning. Whatever you do to one side of an equation, you do to the other. That is the entire law. Everything else in this unit is bookkeeping about direction, distance, and which answers survive. Two things complicate it, and both get their own error entries. Multiplying an inequality by a negative number obeys the law and still reverses the symbol — which is the one place inequalities part company with equations. And clearing fractions out of an equation can manufacture answers that solve the cleared version and fail the original. Those have to be caught, which is why checking is not optional here in the way it usually is. By the end you will be able to do five things. One: find intercepts, and measure distance and midpoint between two points. Two: say why every rational-equation answer is checked in the original. Three: write a line's equation from two points, in the right order. Four: say which move reverses an inequality, and count the reversals. Five: unpack both kinds of absolute value inequality into the right shape.

Three entries setting up where the rest of the course happens. The plane itself, what a graph actually is, and two points worth naming — with a small precision about intercepts that matters more than it seems.

The plane.

Alg 121 The Cartesian Coordinate System

A horizontal axis crossing a vertical one, with every point in the plane located by a pair of numbers in a fixed order.

What a graph is.

Alg 122 Graphing by Plotting Points

The collection of all points whose coordinates satisfy the equation. Plotting enough of them shows the shape — the graph is the whole set, and plotting is how you see it.

The two points worth naming.

Alg 124 Intercepts

Where the graph meets each axis, found by setting the other coordinate to zero. And each one is a point rather than a single number — an answer given as one number has left half of itself out.

Two formulas, and the first is worth recognising rather than memorising — it is the Pythagorean theorem with the legs written as coordinate differences. The second is a plain average, and the two get confused because they look similar.

Measuring between two points.

Alg 126 The Distance Formula

The horizontal gap and the vertical gap are the two legs of a right triangle, and the distance is the hypotenuse. Note where the radical sits — it covers the entire sum, not one term of it.

Finding the halfway point.

Alg 127 The Midpoint Formula

Average the two horizontal coordinates and average the two vertical ones. This one adds and divides where the previous one subtracts and squares — which is the distinction to hold.

Four entries. What the object is, what counts as having solved it, the procedure, and then something the procedure can reveal — that not every equation has the kind of answer you were expecting.

The object.

Alg 131 Linear Equations in One Variable

An equation where the variable never carries a power above one. The word names the degree — it says nothing about how long or short the equation is.

What solving means.

Alg 132 The Solution of an Equation

A value making the equation true, and solving means finding all of them. Confirmation comes from substituting back — never from confidence that the steps were done correctly.

The procedure.

Alg 133 Solving a Linear Equation

Clear the brackets, collect like terms, and apply the opposite of whatever was done to the variable. And the operation goes to both sides — applied to one side only, you have produced a different equation with different answers.

And three possible outcomes.

Alg 135 Identity, Conditional, and Inconsistent Equations

Some equations are true for every value, some for particular values, and some for none at all. The procedure tells you which by what it collapses to — the vanishing variable is information rather than a mistake.

Fractions in an equation get cleared, and clearing is where this unit's characteristic problem appears. Multiplying both sides by something containing the variable can create answers that satisfy the new equation and break the old one.

The object, and the move.

Alg 137 Rational Equations

An equation containing at least one polynomial fraction. Multiplying both sides by the common denominator removes them — and it has to reach every term on both sides, not only the fractional ones.

What clearing can produce.

Alg 138 Checking in the Original Equation

A value the method hands you that does not satisfy the equation you started with. Every answer here gets tested against the original — checking against the cleared version cannot detect the problem, because the cleared version is where it came from.

The procedure.

Alg 140 Solving a Rational Equation

Note which values would make a denominator zero, clear the fractions, solve, and test every answer against the original. Write the excluded values down before clearing — afterwards the denominators are gone and so is the evidence.

Two entries introducing the number that describes a line's steepness. The formula has one condition attached, and it is easy to break without noticing because both halves have to be done the same way round.

The idea.

Alg 142 The Slope of a Line

How much the line climbs against how far it travels, measured between any two of its points — and any two give the same answer.

The formula.

Alg 143 The Slope Formula

The difference in heights over the difference in horizontal positions. Whichever point you subtract first upstairs, subtract first downstairs too — reversing one and not the other flips the sign.

Two ways of writing a line, doing different jobs. One is for reading a line's properties straight off. The other is for constructing a line from what you know — and the procedure that uses it has a required order.

The form for reading.

Alg 145 Slope-Intercept Form

The multiplier on the variable is the slope, and the constant is the height where the line crosses the vertical axis. That constant is a coordinate — the intercept itself is a point.

The form for building.

Alg 146 The Point-Slope Formula

A slope and any single point on the line produce its equation directly. The coordinates being subtracted are the ones from the point you know.

Two points to an equation.

Alg 148 Writing a Line from Two Points

Compute the slope from the two points, then feed that slope and either one of them into the building form. The order is forced — the building form needs a slope, so it cannot come first.

Two relations between lines, both stated as conditions on slopes — and then the pair of lines those conditions cannot describe, because one of them has no slope at all.

The first relation.

Alg 150 Parallel Lines

The same steepness with different crossing points, so they never meet. Both conditions are needed — matching slope and matching intercept describe one line written twice.

The second.

Alg 151 Perpendicular Lines

Lines crossing at a right angle, with slopes that multiply to negative one — each the other's reciprocal with the sign reversed. Vertical lines escape this entirely, and their perpendiculars are horizontal.

And the two that break the rules.

Alg 153 Horizontal and Vertical Lines

A flat line has slope zero and an equation naming a single height. An upright line has no slope at all — the formula would divide by zero — and its equation names a single horizontal position.

Two entries before the inequalities, because the answers change shape here. Up to now an answer has been a number. From this point an answer is a set of numbers, and it needs its own notation.

The notation.

Alg 156 Interval Notation

A stretch of numbers written by its two ends, with square brackets taking the endpoint in and round ones leaving it out. Infinity always takes a round bracket — it is not a number that can be included.

What an answer looks like now.

Alg 158 Solutions of Linear Inequalities

Every value making the statement true, which is usually a continuous stretch rather than one number. Solving means naming that entire set — a single value is not an answer to this kind of question.

Two properties, and this is the whole reason inequalities are taught separately. One behaves exactly like its equation counterpart. The other obeys the same law and still flips the symbol, which is the source of the most common error in the unit.

The one that behaves.

Alg 159 The Addition Property of Inequality

Adding or subtracting the same amount on both sides leaves the symbol pointing the way it did. Direction is the thing being protected throughout.

The one that does not.

Alg 160 The Multiplication Property of Inequality

Multiplying or dividing by a positive number keeps the direction; by a negative number it reverses. The law is still being obeyed — both sides received the same treatment — and the symbol still turns.

The error.

Alg 161 The Kept-Sign Error

Dividing by a negative and leaving the symbol as it was. Nothing about the arithmetic looks wrong, and the answer set describes the opposite of the truth.

The procedure.

Alg 163 Solving a Linear Inequality

The same moves as an equation, with one extra law watching for negative multipliers and divisors. Count them as you go — each one turns the symbol, so two of them turn it back, and losing count is how the error happens.

Two entries. The first chains two conditions around a single expression. The second supplies a reading of absolute value that makes the next topic follow rather than needing to be memorised.

Two conditions at once.

Alg 165 Compound Inequalities

One expression caught between two bounds. Every move applies to all three parts at the same time — whatever reaches the middle reaches both ends.

The reading that explains the rest.

Alg 167 Absolute Value as Distance

The bars around a difference measure how far apart two numbers sit. Once absolute value is read as distance, statements about distance turn directly into these expressions — and the next topic stops needing memorisation.

Two shapes, and the distance reading from the last topic makes both of them obvious. Closer than a given amount traps the expression between two bounds. Farther than it sends the expression out to either side.

Closer than.

Alg 168 Absolute Value Inequalities: Less Than

A distance from zero smaller than some positive amount means the expression sits between the negative and positive versions of it — one interval, and one statement covering both ends.

Farther than.

Alg 169 Absolute Value Inequalities: Greater Than

A distance larger than that amount means the expression lies past it in one direction or the other — two separate stretches, joined by an or rather than trapped between bounds.

And the error.

Alg 170 The Swapped-Cases Error

Solving the closer-than kind as two separate rays, or the farther-than kind as a single trapped interval. The distance reading is the protection — ask whether the expression is being kept near zero or pushed away from it.

The procedure.

Alg 172 Solving an Absolute Value Inequality

Get the absolute value alone on its side, unpack by whichever pattern applies, solve the resulting pieces, and write the answer as intervals. Isolating comes first — a coefficient still sitting outside the bars changes what the bounds mean.

The unit closes on the law the opening quoted, now with everything behind it — and it names what the rest of the unit was, which is a claim about organisation rather than any new mathematics.

The capstone. One law, and bookkeeping.

Alg 180 The Line and the Balance

Whatever is done to one side is done to the other, and that single law drives every procedure in the unit. Everything else was bookkeeping — about which direction a symbol points, about how far apart two things are, and about which answers survive a check.

Five things you can now do. You can find where a graph meets each axis by setting the other coordinate to zero, and measure between two points with Pythagoras wearing coordinate labels. You can say why rational-equation answers get checked in the original: clearing denominators can manufacture values that satisfy the cleared equation and fail the one you started with. You can write a line from two points in the forced order — slope first, because the building form needs one before it can be used. You can say which move reverses an inequality, and you know to count them, because two reversals restore the original direction. And you can unpack both absolute value inequalities by reading the bars as distance — closer than traps, farther than banishes.