Section 16, easy algebra

Inequalities, Modulus & Max-Min

Treat this chapter as number-line thinking: a modulus is a distance, a quadratic inequality is “between or outside the roots”, and AM ≥ GM answers most max-min questions. In the exam, take only the direct versions; the harder ones are a time sink.

How CAT asks it

Count the integer solutions of a linear or quadratic inequality; solve a modulus equation; minimise a sum of distances; find the minimum of x + k/x or the maximum product for a fixed sum; an inequality inside a word problem.

Concepts you must know

  • Multiplying or dividing an inequality by a negative number flips its direction.
  • |x| = x if x ≥ 0 and −x if x < 0. |x − a| is the distance between x and a on the number line.
  • |x − a| < b means a − b < x < a + b. |x − a| > b means x < a − b or x > a + b.
  • With a < b: (x − a)(x − b) < 0 means a < x < b, and (x − a)(x − b) > 0 means x < a or x > b.
  • |x − a| + |x − b| is smallest, equal to |a − b|, for any x between a and b. For three points a < b < c, |x − a| + |x − b| + |x − c| is smallest at the middle point x = b.
  • For positive numbers, a fixed sum gives the largest product when the parts are equal, and a fixed product gives the smallest sum when they are equal. So x + 1/x ≥ 2 for x > 0.
  • The AM–GM inequality behind those rules:
a+b2≥ab(a,b>0; equality when a=b)\frac{a+b}{2} \ge \sqrt{ab} \qquad (a, b > 0;\ \text{equality when } a = b)

Thought process

  1. Move everything to one side and factorise.
  2. Mark the critical points (roots, and where each modulus is zero) on a number line.
  3. Test one value in each region, or use the “between or outside” rule.
  4. When counting, list the integers in the final range and watch strict (<) against non-strict (≤) signs.
  5. For max/min with positive terms, look for a fixed sum or product and apply AM–GM.

Patterns CAT repeats

Numbered so your error log can point at them: “Pattern 2”.

  1. 1Counting integer solutions of linear or quadratic inequalities.
  2. 2Modulus equations, by cases or by distance.
  3. 3Modulus inequalities turned into ranges.
  4. 4Minimum of a sum of distances.
  5. 5AM–GM: the minimum of ax + b/x, or the maximum product for a fixed sum.
  6. 6Inequalities inside word problems, such as the point where one plan becomes cheaper.

Worked examples

Attempt each one before opening its solution, then compare methods.

  1. Example 1

    How many integers satisfy x² − 7x + 10 < 0?

    Show solution

    (x − 2)(x − 5) < 0 means 2 < x < 5, so x = 3 or 4. Answer: 2.

  2. Example 2

    How many integers satisfy |2x − 5| ≤ 7?

    Show solution

    −7 ≤ 2x − 5 ≤ 7, so −2 ≤ 2x ≤ 12 and −1 ≤ x ≤ 6. Answer: 8.

  3. Example 3

    Solve |x − 3| + |x + 2| = 7.

    Show solution

    The points −2 and 3 are 5 apart, so any x between them gives a sum of 5. Moving t beyond either end adds 2t, so 5 + 2t = 7 and t = 1. Check: |1| + |6| = 7 and |−6| + |−1| = 7. Answer: x = 4 or x = −3.

  4. Example 4

    Find the minimum of |x − 1| + |x − 4| + |x − 10|.

    Show solution

    Take the middle point x = 4: 3 + 0 + 6. Answer: 9.

  5. Example 5

    Find the minimum of 4x + 9/x for x > 0.

    Show solution

    The product 4x × 9/x = 36 is fixed, so the sum is smallest when 4x = 9/x = 6, at x = 1.5. Answer: 12.

  6. Example 6

    Two positive integers add to 21. What is their largest possible product?

    Show solution

    Make them as equal as possible: 10 and 11. Answer: 110.

  7. Example 7, word problem

    One phone plan costs ₹200 a month plus ₹0.50 a minute; another costs a flat ₹500. Above how many minutes is the flat plan cheaper?

    Show solution

    200 + 0.5m > 500 gives m > 600. Answer: above 600 minutes.

Traps

  • Forgetting to flip the sign when multiplying by a negative.
  • Mixing up strict and non-strict endpoints when counting integers.
  • Applying AM–GM to terms that can be negative.

Practice set

Timed, about 2.5 minutes per question. Log every miss in the error log.

  1. 1

    How many integers x satisfy −3 < 2x + 1 ≤ 9?

    Show answer

    6: −2 < x ≤ 4.

  2. 2

    Solve |x − 4| = 2x + 1.

    Show answer

    x = 1: the case x ≥ 4 gives x = −5, which contradicts it; check |−3| = 3 = 2 + 1.

  3. 3

    How many integers satisfy x² − 9x + 14 ≤ 0?

    Show answer

    6: 2 ≤ x ≤ 7.

  4. 4

    Find the minimum of x + 16/x for x > 0.

    Show answer

    8, at x = 4.

  5. 5

    A rectangle has a perimeter of 40 m. What is its largest possible area?

    Show answer

    100 m²: a 10 m square.

  6. 6

    Find the minimum value of |x − 2| + |x − 9|.

    Show answer

    7: the distance from 2 to 9.

Log a missFind this chapter’s real CAT questions