🔤 Algebra — Problem Types

Master the common problem patterns and systematic solution approaches for algebra problems.

Basic Equation Problems

One-Step Equations

Recognition: Single operation required to solve Template:

1. Identify the operation
2. Use inverse operation
3. Check solution

Example: Solve $x + 5 = 12$

1. Operation: Addition of 5
2. Inverse: Subtract 5 from both sides
3. Solution: $x = 7$
4. Check: $7 + 5 = 12$ ✓

Common variations:

• Addition: $x + a = b$
• Subtraction: $x - a = b$
• Multiplication: $ax = b$
• Division: $\frac{x}{a} = b$

Two-Step Equations

Recognition: Two operations required to solve Template:

1. Undo addition/subtraction first
2. Undo multiplication/division second
3. Check solution

Example: Solve $2x + 3 = 11$

1. Step 1: Subtract 3: $2x = 8$
2. Step 2: Divide by 2: $x = 4$
3. Check: $2(4) + 3 = 11$ ✓

Common variations:

• $ax + b = c$
• $ax - b = c$
• $\frac{x}{a} + b = c$
• $\frac{x}{a} - b = c$

Multi-Step Equations

Recognition: Multiple operations and simplification required Template:

1. Simplify each side
2. Use inverse operations
3. Check solution

Example: Solve $3(x + 2) - 4 = 8$

1. Simplify: $3x + 6 - 4 = 8$, so $3x + 2 = 8$
2. Solve: $3x = 6$, so $x = 2$
3. Check: $3(2 + 2) - 4 = 3(4) - 4 = 12 - 4 = 8$ ✓

System of Equations Problems

Substitution Method

Recognition: One equation can be easily solved for one variable Template:

1. Solve one equation for one variable
2. Substitute into other equation
3. Solve for remaining variable
4. Find other variable
5. Check solution

Example: Solve $\begin{cases} x + y = 5 \ 2x - y = 1 \end{cases}$

1. Solve first: $y = 5 - x$
2. Substitute: $2x - (5 - x) = 1$
3. Simplify: $2x - 5 + x = 1$, so $3x = 6$, so $x = 2$
4. Find y: $y = 5 - 2 = 3$
5. Check: $2 + 3 = 5$ and $2(2) - 3 = 1$ ✓

Elimination Method

Recognition: Variables can be eliminated by adding/subtracting equations Template:

1. Align like terms
2. Add or subtract equations to eliminate one variable
3. Solve for remaining variable
4. Find other variable
5. Check solution

Example: Solve $\begin{cases} x + y = 5 \ 2x - y = 1 \end{cases}$

1. Align: Already aligned
2. Add: $(x + y) + (2x - y) = 5 + 1$, so $3x = 6$
3. Solve: $x = 2$
4. Find y: $2 + y = 5$, so $y = 3$
5. Check: $2 + 3 = 5$ and $2(2) - 3 = 1$ ✓

Quadratic Equation Problems

Factoring Method

Recognition: Quadratic can be factored into two binomials Template:

1. Set equation equal to zero
2. Factor into $(x - r_1)(x - r_2) = 0$
3. Set each factor equal to zero
4. Solve for x
5. Check solutions

Example: Solve $x^2 - 5x + 6 = 0$

1. Set to zero: Already set
2. Factor: $(x - 2)(x - 3) = 0$
3. Set factors: $x - 2 = 0$ or $x - 3 = 0$
4. Solve: $x = 2$ or $x = 3$
5. Check: $2^2 - 5(2) + 6 = 0$ and $3^2 - 5(3) + 6 = 0$ ✓

Quadratic Formula Method

Recognition: Quadratic cannot be easily factored Template:

1. Identify a, b, c
2. Substitute into formula
3. Simplify
4. Check solutions

Example: Solve $x^2 - 5x + 6 = 0$

1. Identify: $a = 1$, $b = -5$, $c = 6$
2. Formula: $x = \frac{5 \pm \sqrt{25 - 24}}{2}$
3. Simplify: $x = \frac{5 \pm 1}{2}$
4. Solutions: $x = 3$ or $x = 2$
5. Check: $3^2 - 5(3) + 6 = 0$ and $2^2 - 5(2) + 6 = 0$ ✓

Inequality Problems

Linear Inequalities

Recognition: Single variable inequality Template:

1. Solve like equation
2. Flip inequality if multiplying/dividing by negative
3. Graph solution
4. Check solution

Example: Solve $2x + 3 < 11$

1. Solve: $2x < 8$, so $x < 4$
2. Flip: Not needed (positive coefficient)
3. Graph: Open circle at 4, arrow left
4. Check: $x = 3$ gives $2(3) + 3 = 9 < 11$ ✓

Compound Inequalities

Recognition: Two inequalities connected by “and” or “or” Template:

1. Solve each inequality separately
2. Combine using “and” or “or”
3. Graph solution
4. Check solution

Example: Solve $2 < x + 3 < 7$

1. Solve: $2 < x + 3$ and $x + 3 < 7$
2. Simplify: $-1 < x$ and $x < 4$
3. Combine: $-1 < x < 4$
4. Graph: Open circles at -1 and 4, line between
5. Check: $x = 0$ gives $2 < 3 < 7$ ✓

Function Problems

Function Evaluation

Recognition: Find value of function for given input Template:

1. Substitute value for variable
2. Simplify expression
3. Check answer

Example: If $f(x) = 2x + 3$, find $f(4)$

1. Substitute: $f(4) = 2(4) + 3$
2. Simplify: $f(4) = 8 + 3 = 11$
3. Check: $f(4) = 11$ ✓

Function Composition

Recognition: Find value of one function applied to another Template:

1. Find inner function value
2. Substitute into outer function
3. Simplify
4. Check answer

Example: If $f(x) = 2x + 3$ and $g(x) = x^2$, find $f(g(2))$

1. Inner function: $g(2) = 2^2 = 4$
2. Outer function: $f(4) = 2(4) + 3 = 11$
3. Check: $f(g(2)) = f(4) = 11$ ✓

Word Problems

Age Problems

Recognition: Problems about ages of people Template:

1. Define variables for ages
2. Set up equations based on relationships
3. Solve system
4. Check answer

Example: “John is 5 years older than Mary. In 3 years, John will be twice as old as Mary.”

1. Variables: Let $x$ = Mary’s age now
2. Equations: John now = $x + 5$, In 3 years: Mary = $x + 3$, John = $x + 8$
3. Equation: $x + 8 = 2(x + 3)$
4. Solve: $x + 8 = 2x + 6$, so $x = 2$
5. Answer: Mary is 2, John is 7

Distance Problems

Recognition: Problems involving distance, rate, and time Template:

1. Use $d = rt$ formula
2. Set up equations based on given information
3. Solve system
4. Check answer

Example: “A car travels 60 mph for 2 hours, then 40 mph for 3 hours. What is the total distance?”

1. Distance 1: $d_1 = 60 \times 2 = 120$ miles
2. Distance 2: $d_2 = 40 \times 3 = 120$ miles
3. Total: $d = d_1 + d_2 = 120 + 120 = 240$ miles
4. Check: $240 = 60 \times 2 + 40 \times 3$ ✓

Mixture Problems

Recognition: Problems about mixing different concentrations Template:

1. Define variables for amounts
2. Set up equations based on concentrations
3. Solve system
4. Check answer

Example: “Mix 2 liters of 20% acid with 3 liters of 80% acid. What is the concentration?”

1. Variables: Let $x$ = final concentration
2. Equation: $2(0.20) + 3(0.80) = 5x$
3. Solve: $0.4 + 2.4 = 5x$, so $x = 0.56$
4. Answer: 56% acid

Pattern Problems

Arithmetic Sequences

Recognition: Sequence with constant difference Template:

1. Find first term and common difference
2. Use formula $a_n = a_1 + (n-1)d$
3. Calculate desired term
4. Check answer

Example: Find the 10th term of 2, 5, 8, 11, …

1. First term: $a_1 = 2$
2. Common difference: $d = 3$
3. Formula: $a_{10} = 2 + (10-1)(3) = 2 + 27 = 29$
4. Check: $a_4 = 2 + 3(3) = 11$ ✓

Geometric Sequences

Recognition: Sequence with constant ratio Template:

1. Find first term and common ratio
2. Use formula $a_n = a_1 \cdot r^{n-1}$
3. Calculate desired term
4. Check answer

Example: Find the 5th term of 2, 6, 18, 54, …

1. First term: $a_1 = 2$
2. Common ratio: $r = 3$
3. Formula: $a_5 = 2 \cdot 3^{5-1} = 2 \cdot 3^4 = 2 \cdot 81 = 162$
4. Check: $a_4 = 2 \cdot 3^3 = 2 \cdot 27 = 54$ ✓

Common Mistakes and Fixes

Sign Errors

Mistake: $-(x + 3) = -x + 3$ Fix: $-(x + 3) = -x - 3$

Mistake: $2(x - 3) = 2x - 3$ Fix: $2(x - 3) = 2x - 6$

Distribution Errors

Mistake: $3(x + 2) = 3x + 2$ Fix: $3(x + 2) = 3x + 6$

Mistake: $-(x - 3) = -x - 3$ Fix: $-(x - 3) = -x + 3$

Inequality Errors

Mistake: When multiplying by negative, not flipping inequality Fix: Always flip inequality when multiplying/dividing by negative

Mistake: Confusing “and” and “or” in compound inequalities Fix: “And” means both conditions, “or” means either condition

Function Errors

Mistake: $f(2x) = 2f(x)$ Fix: $f(2x) = 2(2x) + 3 = 4x + 3$, not $2(2x + 3) = 4x + 6$

Mistake: $(f \circ g)(x) = f(x) \circ g(x)$ Fix: $(f \circ g)(x) = f(g(x))$, not $f(x) \cdot g(x)$

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