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Helena has saved [tex]\$595[/tex] to put toward snowboarding equipment and lift tickets at the hill. Each lift ticket costs [tex]\$35[/tex]. If she wants to have more than [tex]\$420[/tex] of her savings left to spend on equipment, which inequality represents the number of lift tickets she can buy?

A. [tex]x \ \textless \ 5[/tex]
B. [tex]x \leq 4[/tex]
C. [tex]x \ \textgreater \ 7[/tex]
D. [tex]x \geq 7[/tex]


Sagot :

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]
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