Translating everyday language into algebra helps clarify constraints in budgeting, shopping, and planning. The phrase 3 more than the product of 7 and a number x is less than 26 describes a linear inequality that appears in many simple comparison situations.
Converting the statement into symbols produces 7x + 3
| Phrase Part | Algebraic Expression | Meaning | Example Value |
|---|---|---|---|
| a number x | x | Unknown value we are solving for | 2 |
| the product of 7 and x | 7x | Multiplication of 7 by the unknown | 14 |
| 3 more than the product | 7x + 3 | Add 3 to the previous result | 17 |
| is less than 26 | 7x + 3 | Complete inequality describing the constraint | True for x = 2 |
Solve the Inequality Algebraically
Step by Step Solution
To find acceptable values for x, subtract 3 from both sides to obtain 7x
Any number smaller than 23/7 keeps the original statement true, while numbers equal to or greater than 23/7 make the inequality false.
Interpret in Real World Contexts
Applying the Constraint to Practical Situations
In finance, this inequality might represent a scenario where a base fee of 3 is added to seven times the quantity of items, and the total must remain under a budget of 26.
In scheduling, the expression could describe a workload where fixed setup time combines with variable processing time, and the overall time must stay below a threshold.
Graph the Solution on a Number Line
Visualizing the Range of Acceptable Values
The boundary point at 23/7 is not included, so the graph uses an open circle and shading extends to the left toward negative infinity.
Shading the number line in this direction shows that every tested value in the region satisfies 7x + 3
Test Integer Values Near the Boundary
Check Neighboring Numbers to Confirm the Cutoff
Testing x = 3 gives 7(3) + 3 = 24, which is less than 26, so 3 is acceptable.
Testing x = 4 gives 7(4) + 3 = 31, which is not less than 26, so 4 is not allowed.
Key Takeaways and Recommendations
- Convert phrases into precise algebraic inequalities before solving.
- Use inverse operations carefully, reversing inequality signs only when multiplying or dividing by negatives.
- Verify solutions by testing boundary and nearby values.
- Interpret results in the context of the original problem domain.
FAQ
Reader questions
What does the variable x represent in this statement?
It stands for any unknown number that, when multiplied by 7 and increased by 3, must still keep the total below 26.
Can x be a negative number here?
Yes, negative numbers satisfy the inequality because they produce results well below the limit of 26.
Is x = 23/7 a valid solution?
No, because 23/7 makes the expression equal to 26, and the statement requires a strict less than relation.
How would the inequality change if it were 3 less than the product instead?
It would become 7x - 3 < 26, shifting the boundary point and allowing slightly larger values of x up to 29/7.