1
Solve and justify
Carry a one-variable linear equation to x = a number, naming the operation applied to both sides at each step, and check the answer in the ORIGINAL equation.
Course · Algebra
Linear Equations and Inequalities
Find the value that makes a statement true — and know why each move you made kept it true. Linear equations, literal formulas, inequalities, compound and absolute value statements, and the notation for writing a solution set down.
a = b ⟹ a + c = b + c and the same for ×, ÷ by a nonzero c
multiply an inequality by a negative ⟹ reverse it
|x| = k ⟹ x = k or x = −k two equations, not one
0 = 5 no solution 0 = 0 every number
What this course is about
- Subject
- Algebra
- Lessons
- 13
- Format
- Interactive
Overview
What you will be able to do
By the end you can solve the one-variable real linear equations and inequalities taught here, justify every move, and state the resulting solution set exactly.
2
Recognise the three outcomes
Tell a conditional equation from an identity and from a contradiction, and read 0 = 0 and 0 = 5 correctly rather than as "x = 0".
3
Rearrange a linear formula
Solve a literal equation that is linear in the requested letter, including when that letter appears on both sides, and state when the final divisor is nonzero.
4
Handle inequalities and absolute value
Solve linear, compound and absolute-value statements, reverse the sign when and only when you must, and write the answer in interval notation.
Lessons
13 lessons
Equations come first, then their applications, and then inequalities, absolute value and the notation for a solution set.
What It Means to Solve an EquationA solution is a value that makes the statement true.Not marked
One- and Two-Step EquationsUndoing operations in the reverse of the order they were applied.Not marked
Variables on Both SidesCollecting the unknown on one side without losing a sign.Not marked
Equations with FractionsClearing denominators with the least common denominator.Not marked
Literal Equations and FormulasSolving for one letter among several.Not marked
Identities and ContradictionsWhen every number works, and when none does.Not marked
Ratio, Proportion and PercentThree applications that are one equation in disguise.Not marked
Modelling with Linear EquationsFrom a described situation to an equation you can solve.Not marked
Linear InequalitiesSolving, and the one rule that reverses the sign.Not marked
Compound InequalitiesAnd and or, intersection and union.Not marked
Absolute Value EquationsTwo cases, and when there are none.Not marked
Absolute Value InequalitiesLess-than gives an interval, greater-than gives two.Not marked
Interval and Set-Builder NotationWriting a solution set so a reader knows exactly what it holds.Not marked
Recommended background
Expressions, exponents, the distributive law.
Practice suggestions
After the complete example, cover its answer and do the faded rehearsal before the quiz. The supplied first move is guidance; the remaining moves and the substitution check are yours.
Do the fraction lessons without decimals. x = 7/3 is the answer; x = 2.333 is a rounding of it, and the rest of this path expects the fraction.
In the inequality lessons, test a number from your own answer against the original statement. A reversed sign is invisible in the algebra and obvious in the test.
What it does not cover
Quadratic and higher equations. x^2 = 9 waits for Quadratics and Complex Numbers, where the two roots and the reason there are exactly two are both handled properly.
Systems of several equations at once, which need their own machinery and get it in Systems and Matrices.
Inequalities involving products or quotients of factors, which need sign analysis and arrive in Rational and Radical Expressions and Quadratics and Complex Numbers.