Learning Objectives
Distinguish simplifying an expression from solving an equation.
Solve and check equations and inequalities.
Translate a short situation into algebra and interpret the answer.
Prerequisites: the four introductory algebra lessons in this section. Keep each line of working connected to the previous one. Use a final sentence when the question describes a real quantity.
1. Decide What the Task Requires
Instruction | What to produce |
|---|---|
Simplify | An equivalent expression |
Evaluate | A number after substitution |
Solve an equation | Values making the equality true |
Solve an inequality | A set of allowed values |
Model a situation | A relationship with defined variables and units |
For example, simplifying 3x + 2x gives 5x, not x = 5. There is no equation to solve until an equality is provided. If x = 4 is given, evaluating the same expression gives 20.
2. Connect Distribution and Solving
Worked Example: A Grouped Quantity
Solve 3(x + 2) = 21. One method divides both sides by 3 first, giving x + 2 = 7, then subtracts 2 to obtain x = 5.
Another method distributes first: 3x + 6 = 21. Subtract 6 and divide by 3 to obtain x = 5 again.
Both methods preserve equality. Checking in the original equation gives 3(5 + 2) = 21. Choose a valid method that keeps the working clear.
3. Use a Check That Fits the Question
For an equation, substitute the proposed solution. For an inequality, check the boundary and a value from the included region. For a word problem, also check the unit and whether the value is possible in the situation.
Common Mistakes
An equal sign should connect quantities with the same value; do not use it merely to mean “next step.”
Include a term's sign when distributing and combining.
A solution set such as x > 2 is not the single value 2, and the strict boundary is excluded.
4. Mixed Practice
1. Simplify 4(2x − 1) − 3x + 6.
2. Evaluate 3a² + 2a when a = −2.
3. Solve 5x + 4 = 29.
4. Solve 9 − 2x > 15.
5. A number is multiplied by 4, then 7 is added. The result is 35. Find the number.
6. A $6 entry fee and $3 per activity must cost at most $23. Find the maximum whole number of activities.
7. A rectangle has width w cm and length w + 4 cm. Its perimeter is 32 cm. Find both dimensions and its area.
Worked Solutions
1. Distribute to get 8x − 4 − 3x + 6. Combine like terms to obtain 5x + 2.
2. Substitute with parentheses: 3(−2)² + 2(−2) = 12 − 4 = 8.
3. Subtract 4 to get 5x = 25, then divide by 5. The solution is x = 5, and 5(5) + 4 = 29 checks.
4. Subtract 9: −2x > 6. Divide by −2 and reverse the symbol: x < −3. The boundary −3 gives equality, so it is excluded.
5. Write 4n + 7 = 35. Subtract 7 and divide by 4 to get n = 7.
6. Write 6 + 3n ≤ 23. Then n ≤ 17/3, so the maximum whole number is 5. Five activities cost $21; six cost $24 and are not allowed.
7. Use 2w + 2(w + 4) = 32. This gives 4w + 8 = 32, so w = 6 cm and the length is 10 cm. The area is 60 cm²; the perimeter check is 2(6 + 10) = 32 cm.
5. Summary
- Read the instruction before choosing an algebraic method.
- Preserve equality or order at every step.
- Check answers in the original statement.
- For a context, give the quantity, unit, and any whole-number restriction.