AMC 10 · 2010 · #11

Grade 7 algebra

Archetype Coordinate Plane Figure Computation └ Sub-archetype Figure Constraint Equations for Unknown Quantity

interval-arithmeticlinear-equations-one-var convert-to-algebra ↑ Prerequisites: interval-arithmetic

📏 Short solution 💡 2 insights

Problem

For the compound inequality a ≤ 2x + 3 ≤ b, the set of x that make it true is an interval of length 10. Find b - a.

Pick an answer.

(A)

$\ 6$

(B)

$\ 10$

(C)

$\ 15$

(D)

$\ 20$

(E)

$\ 30$

AMC 10 2010 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Introduce a Variable

Tool #4 (Introduce a Variable): the inequality traps x between two bounds, so isolate x by doing the same steps to all three parts — subtract 3, then divide by 2 — turning the given bounds into plain bounds on x. Tool #7 (Subproblems): split the job into (i) find the two endpoints of the x-interval, then (ii) subtract them to get the length. Tool #11 (Work Backwards): the length 10 is the finished result; write the length as an expression in b - a and reverse the division-by-2 to recover b - a.

1STEP 1

Isolate x in all three parts

Subtract 3 from all three parts, then divide all three by 2 — a positive divisor keeps the signs — giving $\frac{a-3}{2} \le x \le \frac{b-3}{2}$ .

a ≤ 2x+3 ≤ b → a-3 ≤ 2x ≤ b-3 → (a-3)/2 ≤ x ≤ (b-3)/2

2STEP 2

Measure the interval length

Length is the upper endpoint minus the lower one; the two -3 terms cancel and leave $\frac{b-a}{2}$ .

length = (b-3)/2 - (a-3)/2 = (b-3)-(a-3)/2 = (b-a)/2

3STEP 3

Set the length to 10 and reverse it

That length is given as 10, so $\frac{b-a}{2} = 10$ ; multiplying both sides by 2 undoes the halving and gives b - a = 20, choice (D).

(b-a)/2 = 10 → b-a = 10 × 2 = 20

Answer

Check with concrete numbers: pick the x-interval 0 ≤ x ≤ 10 (length 10). Then 2x+3 runs from 2(0)+3 = 3 up to 2(10)+3 = 23, so a = 3, b = 23, and b - a = 20. This matches, and it shows the choice of interval does not matter — the slope 2 always doubles the gap. Choice (B) 10 would be the answer only if the middle were plain x; the × 2 is exactly why the gap grows to 20, so (D) is right.

💡Key takeaway

Solving for x divides the gap by the number in front of x, so a slope of 2 makes the bound-gap b-a twice the length of the x-interval.

Isolate x in all three parts
Measure the interval length
Set the length to 10 and reverse it

Isolate x in all three parts

Measure the interval length

Set the length to 10 and reverse it

More like this