08/26/25

An expression is a mathematical phrase that contains numbers, variables, and operations but no equal sign. Its only goal is to be simplified by combining like terms if possible.
An equation contains an equal sign and expresses a relationship between two sides; its goal is to solve for the unknown variable.
Key idea: if there is an equal sign, it’s an equation and you should try to solve for the variable; if there is no equal sign, it’s just an expression and you simplify.

Expression in part a: no equals sign present. Simplify by combining like terms.
- Given: 5x − 4x + 7 + 3
- Simplified result: x + 10
Expression in part b: no equals sign present. Simplify by combining like terms.
- Given: −7m + 8m + 15 − 2
- Simplified result: m + 13
Equation in part c: has an equals sign. Solve for the variable.
- Given: 5x + 7 - 4x = 3
- Left side simplifies to x + 7, so the equation is x + 7 = 3 → x = -4
- Note: solution set notation is used; the solution is the set { -4 } (not the format "x = -4" in some systems).
Equation in part d: has an equals sign. Solve for the variable.
- Given: -7m + 15 + 8m = -2
- Left side simplifies to m + 15; solved by subtracting 15 from both sides → m = -17

Objective when solving: get the variable by itself on one side of the equal sign.
If a term is connected to the variable by addition or subtraction, use the Addition Property of Equality.
If a term is connected to the variable by multiplication or division, use the Multiplication Property of Equality.
It’s fine to have the variable on either side of the equation; the goal is to isolate the variable with consistent operations on both sides.
General workflow:
- 1) Check for parentheses; if present, consider distributing to clear parentheses (Distributive Property).
- 2) Combine like terms on each side as needed.
- 3) Move terms involving the variable to one side and constants to the other side by adding/subtracting opposites.
- 4) If the variable has a coefficient besides 1, divide (or multiply by the reciprocal) to isolate the variable.
- 5) Check your solution by substituting back into the original equation.
Important caveat: never divide by zero; division by zero is undefined.
Two typical outcomes for linear equations:
- A unique solution (one value of the variable).
- No solution (empty set, symbolized by ∅).
- Also, sometimes every real number is a solution (when both sides are identical, e.g., 4x = 4x).

Example: Solve 5x + 7 - 4x = 3
- Combine like terms on the left: (5x - 4x) + 7 = 3 \ \ x + 7 = 3
- Subtract 7 from both sides (Addition Property of Equality with the opposite): x = -4
- Solution set: { -4 }
Example: Solve -7m + 15 + 8m = -2
- Combine like terms on the left: (-7m + 8m) + 15 = -2 \ m + 15 = -2
- Subtract 15 from both sides: m = -17
Understanding properties in context
- For the equation 7t - 6t + 3 = 3
- Combine like terms: t + 3 = 3
- Subtract 3 from both sides: t = 0
- For a scenario with a variable on both sides, e.g., -12x - 9 = 11 - 7x
- Move the variable terms to one side (add 7x to both sides or add 12x to both sides): using addition on both sides gives
  -12x - 9 + 7x = 11 or -12x - 9 + 12x = 11 - 7x ; common clean path: add 12x to both sides -> -9 = 11 + 5x ; subtract 11: -20 = 5x ; divide by 5: x = -4
Distinguishing formatting in some systems
- Some software wants the solution in a specific format (e.g., not "x = -4" but a single value in the solution set). When in doubt, report the numerical solution and the corresponding set notation.

Use a calculator to avoid arithmetic mistakes, especially with negatives.
Example: evaluate expressions and verify by substituting back.
For division, be mindful of the rule: Zero cannot be a divisor. If you see an expression like rac{0}{15}, the result is 0. If you see rac{15}{0}, the operation is undefined (Error).

If there is a parenthesis, clear it first by distributing.
- Example: -3\,(x + 5) + 15 = 9
- Distribute: (-3x - 15) + 15 = 9
- Combine like terms: -3x = 9
- Solve: divide by -3: x = -3
Important note: the order matters for readability, but any correct sequence of valid steps that leads to the same solution is acceptable as long as you stay consistent.

Basics: The square root operation is the inverse of squaring. For a perfect square, the square root is an integer.
- \sqrt{4} = 2, \quad \sqrt{9} = 3, \quad \sqrt{16} = 4, \quad \sqrt{25} = 5, \quad \sqrt{36} = 6.
Negative in front of a radical remains outside: -\sqrt{4} = -2.
When the radicand is not a perfect square, simplify using prime factorization:
- Goal: express the radicand as a product of a perfect square and the smallest leftover radical.
- Method: Prime factorization into two columns (left primes that divide evenly, right the remaining factor). Extract pairs from under the radical and keep the unpaired factors inside.
Worked examples from the transcript:
- (\sqrt{36} = 6) and (-\sqrt{4} = -2).
- (\sqrt{32}): 32 = 2^5 → pairs of 2 give 2^2 = 4 outside; left inside is 2. Result: \sqrt{32} = 4\sqrt{2}.
- (\sqrt{90}): 90 = 2 × 3^2 × 5. Pairs of 3 give outside a factor of 3; inside left is 2 × 5 = 10. Result: \sqrt{90} = 3\sqrt{10}.
- (\sqrt{54}): 54 = 2 × 3^3. Pairs of 3 give outside a factor of 3; inside left is 2 × 3 = 6. Result: \sqrt{54} = 3\sqrt{6}.
Practical tips:
- If a radicand contains a visible perfect square factor, pull it out.
- If not, use the prime factorization method (column method or tree method).
- If a result is decimal (non-perfect square), calculators will show a decimal approximation; radicals provide exact answers.

Multiple methods can solve many problems; the goal is consistent correctness, not a single “right” method.
Academic integrity: do not cheat; if you can justify your method and arrive at the right answer, you’re on the right track.
Real-world relevance: solving equations and simplifying radicals are foundational skills in science, engineering, economics, and data analysis.

Distinguish expressions vs equations and identify the appropriate operation (simplify vs solve).
Use Addition Property of Equality to remove terms added to the variable side; use Multiplication Property of Equality to remove terms multiplied to the variable side.
When solving, consider moving the variable to one side and constants to the other, then isolate the variable by division or multiplication.
Apply distributive property to clear parentheses before solving when needed.
For radicals, use prime factorization to simplify to the form a\sqrt{b} where b is square-free.
Remember special cases:
- Infinite solutions: e.g., 4x = 4x
- No solution: e.g., 4x+2 = 4x+3 (empty set, denoted by \emptyset)
Always verify by substitution when possible.