Depth · one concept from first principles · step 2 of 4
Opposite signs turn |b|=2|a| into b=-2a
Two points A(a) and B(b) satisfy 2|a| = |b|, ab ＜ 0 and a + b = 4. Find the midpoint of A and B, then the midpoint of that point and A.
Number system
Your answer
How to solve
Strategy Convert to Algebra — The absolute-value condition alone allows b = 2a or b = -2a. The sign condition picks one, which turns the problem into a one-variable equation. After finding A and B, apply the midpoint rule twice on the number line.
1STEP 1
Decide the signs
A negative product means a and b have opposite signs. |b| = 2|a| says b is twice as far from 0 as a, and on the opposite side, so b = -2a.
a × b ＜ 0, |b| = 2|a| → b = -2a
b = 2a would give a and b the same sign, which is ruled out.
7.NS.A.2Convert To AlgebraThe conditions force .
Why?
Like signs multiply to a positive and unlike signs to a negative, so the negative product means and have opposite signs.
🧱Signs of products and quotientsMultiplying by a negative number flips a number to the other side of zero, so two flips land back on the positive side: for two nonzero numbers, a product or quotient with like signs is positive and one with unlike signs is negative (zero itself has no sign).
Why?
On the other side of from and twice as far out, must be .
Why?
says sits twice as far from as does.
🧱Absolute value is distance from zero|a| is how far a sits from 0 on the number line, whichever side it is on, so it is never negative and |a - b| is the gap between a and b.
Why?
Flipping the number line over sends each point to its negative at the same distance from , so the point on the other side of that is twice as far as is .
🧱Rigid motion preserves length and angleSliding, turning, or flipping a shape lays it exactly onto a copy, so every length and angle stays the same (folding is the flip case).
2STEP 2
Use the sum
Substitute b = -2a into a + b = 4.
a + (-2a) = -a = 4 → a = -4, b = 8
a = -4 and b = 8; check: 2 × 4 = 8, (-4) × 8 ＜ 0, -4 + 8 = 4.
7.NS.A.1Convert To Algebra3STEP 3
First midpoint
A♥B is the average of -4 and 8.
((-4) + 8)/2 = 2
The midpoint of A and B is at 2.
7.NS.A.1Draw A Diagram4STEP 4
Second midpoint
(A♥B)♥A is the midpoint of the point 2 and A(-4).
(2 + (-4))/2 = -1
Halfway between 2 and -4 is -1.
7.NS.A.1Identify SubproblemsAnswer
-1
On the number line, A is at -4 and B at 8, 12 apart, so their midpoint is 6 to the right of A, at 2. The midpoint of -4 and 2 is 3 to the right of A, at -1. C lies a quarter of the way from A to B, as it should.
Takeaway
A sign condition can turn an absolute-value equation with two cases into a single equation.
- Decide the signs
- Use the sum
- First midpoint
- Second midpoint
Where next?
Level upsuggested
Another one like thissuggested
Same look, different logicsuggested
Step back and solidifysuggested
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