Solve 3x + 8 > 2, when x is (i) an integer, (ii) a real number.
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3x + 8 > 2 ⇒ 3x > -6 ⇒ x > -2
(i) Integers: x ∈ {-1, 0, 1, 2, ...} (ii) Real numbers: x ∈ (-2, ∞)
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Exercise 6.1
Solve -15 < 3(x - 2) ≤ 0 and represent the solution on a number line.
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-15 < 3(x - 2) ≤ 0 Divide by 3: -5 < x - 2 ≤ 0 Add 2: -3 < x ≤ 2
Solution set: (-3, 2]. Number line: open circle at -3, closed circle at 2, shaded between.
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Exercise 6.1
Solve 7x + 3 < 5x + 9. Show the graph of the solution on a number line.
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7x + 3 < 5x + 9 2x < 6 x < 3
Solution: (-∞, 3). Open circle at 3 on the number line, shading left.
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Exercise 6.1
Solve the system: 3x - 7 > 5x - 1 and 2x + 5 ≥ 1.
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First: 3x - 7 > 5x - 1 -2x > 6 ⇒ x < -3
Second: 2x + 5 ≥ 1 ⇒ 2x ≥ -4 ⇒ x ≥ -2
No real number satisfies both x < -3 and x ≥ -2. Solution set = ∅
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Exercise 6.1
The cost of a pen is ₹10 and a notebook is ₹20. A student can spend at most ₹100. If he buys x pens and y notebooks, write the inequality representing the constraint.
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Cost = 10x + 20y At most ₹100: 10x + 20y ≤ 100 Dividing by 10: x + 2y ≤ 10 with x ≥ 0, y ≥ 0 (and x, y integers for practical purposes).