autumn is working two summer jobs, making $20 per hour tutoring and making $13 per hour landscaping. in a…

Explanation:

Step1: Calculate tutoring earnings

Autumn tutors 7 hours at $20/hour, so earnings: ( 7 \times 20 = 140 ) dollars.

Step2: Define landscaping hours

Let ( x ) be landscaping hours. Total hours: ( 7 + x \leq 13 ), so ( x \leq 6 ).

Step3: Set up earnings inequality

Total earnings: ( 140 + 13x \geq 190 ). Subtract 140: ( 13x \geq 50 ). Then ( x \geq \frac{50}{13} \approx 3.85 ).

Step4: Find maximum whole ( x )

( x ) must be whole, ( \leq 6 ), and ( \geq 3.85 ). Max whole ( x ) is 6? Wait, check earnings: ( 140 + 13(6) = 140 + 78 = 218 \geq 190 ). Wait, but total hours ( 7 + 6 = 13 ), which is allowed. Wait, but wait, ( \frac{50}{13} \approx 3.85 ), so minimum ( x ) is 4? Wait no, wait: ( 13x \geq 50 ) → ( x \geq 50/13 ≈ 3.846 ), so ( x ) must be at least 4 (since whole hours). Wait, but we need maximum ( x ) such that ( 7 + x \leq 13 ) (so ( x \leq 6 )) and ( 140 + 13x \geq 190 ). Let's check ( x = 6 ): earnings 218 ≥ 190, hours 13 ≤13: valid. ( x = 6 ) is allowed? Wait, but wait, maybe I made a mistake. Wait, the problem says "maximum number of whole hours landscaping" while meeting both total hours ≤13 and earnings ≥190. Let's recheck:

Tutoring: 7h → $140.

Landscaping: x h, $13/h.

Total hours: 7 + x ≤13 ⇒ x ≤6.

Total earnings: 140 +13x ≥190 ⇒13x ≥50 ⇒x ≥50/13≈3.846. So x must be ≥4 (since whole hours) and ≤6. So maximum x is 6? Wait, but 7+6=13 (total hours ok), earnings 140+78=218≥190. So yes, 6. Wait, but wait, maybe I miscalculated. Wait, 50 divided by 13 is approximately 3.846, so the minimum x is 4? No, wait, if x=3, earnings: 140+39=179 <190. x=4: 140+52=192 ≥190. So x must be at least 4. And at most 6 (since 7+6=13). So maximum whole x is 6.

Wait, but let's confirm:

If x=6:

Total hours: 7+6=13 (≤13: ok).

Total earnings: 720 +613=140+78=218 (≥190: ok).

If x=7: total hours 14>13: invalid.

So yes, maximum is 6.

Answer:

6