Equations and Inequalities
What this domain tests
The Equations and inequalities domain tests solving and interpreting solution sets: linear and quadratic equations, rational and radical equations, absolute value, systems, and inequalities. CLEP College Algebra expects both algebraic solutions and correct interval notation, including excluded values.
Linear equations and systems
- Solve linear equations by applying inverse operations to both sides. An identity is true for all values; a contradiction has no solution.
- For systems, substitution or elimination gives an ordered pair. Parallel distinct lines produce no solution; the same line produces infinitely many solutions. Always check the ordered pair in both original equations.
Quadratic equations
- Write in standard form $ax^2+bx+c=0$. Use factoring and the zero product property, the square root property, completing the square, or the quadratic formula.
- The discriminant $b^2-4ac$ determines the type of roots: positive gives two distinct real roots, zero gives one repeated real root, and negative gives two complex conjugate roots.
- For $ax^2+bx+c=0$, the sum of roots is $-b/a$ and the product is $c/a$.
Inequalities and absolute value
- Multiplying or dividing an inequality by a negative number reverses its direction.
- For $|X|<k$, write $-k<X<k$; for $|X|>k$, write $X<-k$ or $X>k$.
- For rational inequalities, find critical values from numerator and denominator, test intervals on a sign chart, and exclude zeros of the denominator. Use a bracket at a numerator zero only if the inequality allows equality and the expression is defined there.
Radical and rational equations
- Squaring both sides can introduce extraneous solutions, so every candidate must be checked in the original equation.
Worked example
Solve $\sqrt{2x+3}=x$.
Square both sides: $2x+3=x^2$, so $x^2-2x-3=0$, which factors as $(x-3)(x+1)=0$. Candidates are $x=3$ and $x=-1$. Check $x=3$: $\sqrt{9}=3$, valid. Check $x=-1$: $\sqrt{1}=1$, but the right side is $-1$, so it is extraneous. The solution set is ${3}$. Squaring erased the sign information that the check restores.
Traps that cost marks
- Forgetting to reverse an inequality after multiplying or dividing by a negative.
- Using a closed interval at a value that makes a denominator zero.
- Dividing both sides by a variable and losing the solution where that variable equals zero.
- Skipping the check after squaring a radical equation.