Double inequalities and systems

A double inequality squeezes an expression between two numbers. Solve it by doing the same thing to all three parts at once.

• –3 < 2x + 5 ≤ 11: subtract 5 from all parts, –8 < 2x ≤ 6; divide by 2, –4 < x ≤ 3.
• 1 ≤ 7 – 2x < 5: subtract 7, –6 ≤ –2x < –2; divide by –2 and reverse both signs, 3 ≥ x > 1, which reads 1 < x ≤ 3.

Check an end point: at x = 3, 7 – 2x = 1, which is allowed (≤); at x = 1, 7 – 2x = 5, which is not allowed (<). That matches 1 < x ≤ 3.

A system of inequalities must be true together. Solve each one separately, mark both on the same number line, and keep only the overlap.

• 2x – 1 > 3 gives x > 2, and 3x + 2 ≤ 20 gives x ≤ 6.
• On the number line: an open circle at 2 shaded to the right, and a filled circle at 6 shaded to the left. They overlap between them.
• Solution: 2 < x ≤ 6, or (2, 6].

If the two shaded parts do not meet, there is no solution. x < 1 and x > 4 can never both be true.

Formulas in a range. To keep a liquid between 15 °C and 30 °C, use F = (9/5)C + 32 on each end: 15 °C is 59 °F and 30 °C is 86 °F, so 59 < F < 86. Going the other way, use C = (5/9)(F – 32) on all three parts.

Mixtures. A tank holds 500 litres of a 10% acid solution. How much 40% acid solution must be added so the mixture is between 20% and 25% acid?

• If x litres are added, the acid is 50 + 0.4x litres in a total of 500 + x litres.
• So 0.20(500 + x) < 50 + 0.4x < 0.25(500 + x).
• Left part: 100 + 0.2x < 50 + 0.4x, so x > 250. Right part: 50 + 0.4x < 125 + 0.25x, so x < 500.
• Answer: between 250 and 500 litres.

A warning: all the inequalities in this chapter have the variable on the top, never in a denominator. With something like 2/x > 1 you cannot simply multiply by x, because you do not know whether x is positive (keep the sign) or negative (reverse it).