Use the explicit arithmetic rule aₙ = a₁ + (n − 1)d. From a₄ = 19 and a₁₀ = 43 there are six steps between the terms, so 6d = 43 − 19 = 24, giving d = 4, and a₁ = 19 − 3·4 = 7. Then aₙ = 7 + 4(n − 1) = 4n + 3. The geometric rule is bₙ = b₁·r^(n−1) = 3·2^(n−1). Compare term by term: n = 1 gives 3 vs 7; n = 2 gives 6 vs 11; n = 3 gives 12 vs 15 (still behind); n = 4 gives 24 vs 19, so the geometric terms first pass the arithmetic terms at n = 4. The tempting wrong answer n = 3 comes from writing bₙ = 3·2ⁿ, multiplying by the ratio n times instead of n − 1 times; that shifts every geometric term one position early and gives 24 at n = 3. The choice n = 5 comes from treating the given a₄ = 19 as the first term, producing aₙ = 4n + 15, whose larger values delay the crossing by one step. The claim that no such n exists comes from checking only the first two or three terms, where the arithmetic values are indeed larger, and stopping: exponential growth with ratio greater than 1 always overtakes linear growth eventually, so a crossover must exist.

An x-intercept is the point where y = 0. Substituting y = 0 into 4x − 5y = 20 gives 4x = 20, so x = 5 and the shared point is (5, 0). Line ℓ therefore passes through (5, 0) and (−3, 7), so by the slope formula m = (7 − 0)/(−3 − 5) = 7/(−8) = −7/8. A y-intercept is where x = 0, so travel from (5, 0) back to x = 0, a run of −5: y = 0 + (−7/8)(0 − 5) = 35/8. Check: the line y = −(7/8)x + 35/8 gives y = 0 at x = 5 and y = 7 at x = −3. ✓ The most tempting wrong choice is −4, which is the y-intercept of 4x − 5y = 20 rather than of ℓ; that line is only used to donate its x-intercept, and its own y-intercept belongs to a different line entirely. Choosing 5 reports the x-intercept when the question asked where the line meets the y-axis. Choosing −35/8 uses a slope of +7/8, dropping the negative even though the line falls from left to right.

📌 The solution is where the lines intersect = (1, 2).
Verify: y = x + 1 → 2 = 1 + 1 ✓
y = −x + 3 → 2 = −1 + 3 ✓

📌 −2x = 6 → x = −3

Slope = (y₂ - y₁)/(x₂ - x₁) = (2 - (-2))/(2 - (-2)) = 4/4 = 1.

📌 Slope-intercept form. m=slope, b=y-intercept.

Subtract 5: 3x > 9. Divide by 3: x > 3.

x/60 + (280 − x)/40 = 5.5. Multiply by 120: 2x + 3(280 − x) = 660 → 840 − x = 660 → x = 180.

📌 Dashed line = strict inequality (< or >, not ≤ or ≥).
Shaded below the line → y < mx + b.
The boundary line has y-intercept 1 and slope 1, so its equation is y = x + 1. The shaded region lies below that dashed boundary, so the inequality is y < x + 1.