To determine the correct inequality expressing the number of lift tickets Helena can buy, let's go through the situation step-by-step.
1. Total Savings and Ticket Costs:
- Helena has [tex]$595 saved.
- Each lift ticket costs $[/tex]35.
2. Savings Requirement:
- Helena wants to have more than [tex]$420 left for equipment.
3. Calculation of Remaining Savings:
- If Helena buys \( x \) lift tickets, the cost is \( 35x \).
- The remaining savings after buying \( x \) lift tickets would be \( 595 - 35x \).
4. Formulating the Inequality:
- We need Helena's remaining savings to be more than $[/tex]420, so:
[tex]\[
595 - 35x > 420
\][/tex]
5. Solve for [tex]\( x \)[/tex]:
- Subtract 420 from both sides of the inequality:
[tex]\[
595 - 35x - 420 > 0
\][/tex]
Simplifying the left-hand side, we get:
[tex]\[
175 - 35x > 0
\][/tex]
- Subtract 175 from both sides:
[tex]\[
175 - 175 - 35x > 0 - 175
\][/tex]
Simplifying, we get:
[tex]\[
-35x > -175
\][/tex]
- Divide both sides by -35, and remember to reverse the inequality sign when dividing by a negative number:
[tex]\[
x < \frac{175}{35}
\][/tex]
Simplifying:
[tex]\[
x < 5
\][/tex]
6. Interpreting the Result:
- This implies that the maximum number of tickets she can buy is 4 because for [tex]\( x < 5 \)[/tex], the integer solutions are [tex]\( x = 0, 1, 2, 3, 4 \)[/tex].
7. Choosing the Correct Inequality:
- Among the given options:
- A. [tex]\( x < 5 \)[/tex]
- B. [tex]\( x \leq 4 \)[/tex]
- C. [tex]\( x > 7 \)[/tex]
- D. [tex]\( x \geq 7 \)[/tex]
- The correct inequality that aligns with our solution is:
[tex]\( x \leq 4 \)[/tex].
Thus, the correct answer is:
[tex]\[ \boxed{B} \, x \leq 4 \][/tex]
