Solve the linear inequality. Express the solution set using set-builder and interval notation. Graph the solution set. [tex]5 x\ \textless \ 35[/tex]
The solution set expressed in set-builder notation is [tex] \{x \mid[/tex] [$\square$\}].
The solution set expressed in interval notation is [$\square$].
Answer (2)
Divide both sides of the inequality by 5: x < 7 .
Express the solution in set-builder notation: { x ∣ x < 7 } .
Express the solution in interval notation: ( − ∞ , 7 ) .
The solution set is all numbers less than 7: ( − ∞ , 7 )
Explanation
Isolating x We are given the inequality 5 x < 35 . To solve for x , we need to isolate x on one side of the inequality. We can do this by dividing both sides of the inequality by 5.
Dividing by 5 Dividing both sides of the inequality 5 x < 35 by 5, we get: 5 5 x < 5 35 x < 7
Set-builder Notation The solution set in set-builder notation is the set of all x such that x is less than 7. This is written as: { x ∣ x < 7 }
Interval Notation The solution set in interval notation is the interval from negative infinity to 7, not including 7. This is written as: ( − ∞ , 7 )
Final Answer Therefore, the solution to the inequality 5 x < 35 is x < 7 . In set-builder notation, the solution set is { x ∣ x < 7 } . In interval notation, the solution set is ( − ∞ , 7 ) .
Examples
Imagine you're saving money for a new bicycle that costs $35. If you save $5 each week, this problem helps you determine how many weeks it will take to save less than the bicycle's cost. By solving the inequality 5 x < 35 , you find that in less than 7 weeks, you'll have saved enough. This concept applies to budgeting, planning expenses, and understanding financial goals.
The solution to the inequality 5 x < 35 is x < 7 . In set-builder notation, this is expressed as { x ∣ x < 7 } , and in interval notation, it is ( − ∞ , 7 ) . Thus, all values of x are less than 7.
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