Operations & Word Problems

Depth · one concept from first principles · step 2 of 4

No solution forces a zero coefficient

Gather the x-terms of ax+1 ＜ 3x+b on one side. Decide what the coefficients must be so that no value of x works, then make a+b as large as possible.
Expressions
Your answer
How to solve
Strategy Convert to Algebra — Write the inequality as (number) · x ＜ (number). A nonzero coefficient always leaves a whole ray of solutions, so rule that out; then the inequality has no x left in it and is simply true or false.
1STEP 1

Collect the x-terms

Subtract 3x and 1 from both sides.

ax+1 ＜ 3x+b → (a-3)x ＜ b-1
2STEP 2

A nonzero coefficient always gives solutions

If a-3 ≠ 0, dividing by a-3 always leaves solutions, whatever its sign, so a-3 must be 0.

a-3=0
3STEP 3

Make the leftover statement false

With a=3 the inequality reads 0 · x ＜ b-1, that is 0 ＜ b-1, for every x. It has no solution exactly when this statement is false.

0 ＜ b-1 is false ⇔ b-1 ≤ 0 ⇔ b ≤ 1
4STEP 4

Maximize a+b

Since a=3 and b ≤ 1, the sum is at most 3+1.

a+b=3+b ≤ 3+1=4
Answer
4
Try a=3, b=1: the inequality becomes 3x+1 ＜ 3x+1, i.e. 1 ＜ 1, false for every x, so there is indeed no solution. With b=1.5 it would read 1 ＜ 1.5, true for every x, so b cannot exceed 1.
Takeaway

"No solution" is a strong clue: the x has to cancel out completely.

  • Collect the x-terms
  • A nonzero coefficient always gives solutions
  • Make the leftover statement false
  • Maximize a+b
Where next?
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▶ Practice — 12 problems