A company makes scientific calculators and graphic calculators.
Each day they make x scientific calculators and y graphic calculators.
These inequalities describe the number of scientific and graphic calculators they make each day:
$x < 180$ $y \leq 90$ $x + y \leq 240$
(a) Complete these two statements:
The company makes fewer than …… scientific calculators each day.
The company can make a maximum of ….. calculators each day.
(b) Scientific calculators cost \($12\) to make.
Graphic calculators cost \($18\) to make.
Each day the company spends at least \($2700\) making calculators.
Show that $2x + 3y \geq 450$.
(c) The region R satisfies these four inequalities:
$x < 180$, $y \leq 90$, $x + y \leq 240$, $2x + 3y \geq 450$
By drawing four suitable lines and shading unwanted regions, find and label the region R.
(d) Scientific calculators are sold for a profit of \($10\).
Graphic calculators are sold for a profit of \($30\).
Calculate the maximum profit made by the company in one day.
▶️ Answer/Explanation
(a) 180 and 240
The first blank comes from the inequality x ≤ 180. The second blank is the maximum total calculators from x + y ≤ 240.
(b) Total cost is 12x + 18y ≥ 2700. Dividing all terms by 6 gives 2x + 3y ≥ 450.
(c) The region R is where all four inequalities are satisfied simultaneously. This involves drawing all four boundary lines and shading the appropriate areas.
(d) \$4200
The maximum profit occurs at the vertex (150, 90). Profit = 10×150 + 30×90 = 1500 + 2700 = \$4200.
(a) 
Write down the inequality shown by the number line.
(b) -3 ≤ 2x + 3 < 9
(i) Solve the inequality.
(ii) Write down all the integer values of x that satisfy the inequality.
(c) Solve the equations.
(i) \( 3(3-x) – \frac{2(x+2)}{5} = 1 \)
(ii) \( \frac{5}{x+3} = \frac{3}{x+5} \)
▶️ Answer/Explanation
(a) -2 < x ≤ 4
The open circle at -2 means x > -2, and the closed circle at 4 means x ≤ 4.
(b)(i) -3 ≤ x < 3
Subtract 3 from all parts: -6 ≤ 2x < 6. Then divide by 2: -3 ≤ x < 3.
(b)(ii) -3, -2, -1, 0, 1, 2
These are all integers between -3 (included) and 3 (not included).
(c)(i) x = 2
Multiply all terms by 5: 45-15x – 2x-4 = 5. Combine like terms: 41-17x = 5.
Then -17x = -36 → x = 36/17 ≈ 2.117 (exact form).
(c)(ii) x = -8
Cross-multiply: 5(x+5) = 3(x+3). Expand: 5x+25 = 3x+9.
Subtract 3x: 2x+25 = 9. Subtract 25: 2x = -16 → x = -8.