Depth · one concept from first principles · step 2 of 4
Match interval endpoints, splitting on the sign
The values of that satisfy the inequality are exactly . Find the integers and . (Assume .)
Number systemExpressions
Your answer
How to solve
Strategy Identify Subproblems — Split on the sign of a (two subproblems). In each case match the two endpoints, which gives two linear equations in a and b; then keep only the case consistent with its sign assumption.
1STEP 1
Remove the absolute value
|ax+1| ≤ b means ax+1 lies between -b and b. Subtract 1 throughout.
-b ≤ ax+1 ≤ b → -b-1 ≤ ax ≤ b-1
An absolute value at most b means the distance from 0 is at most b.
6.NS.C.7Convert To Algebrasays exactly the same thing as .
Why?
is how far sits from on the number line, so it is at most exactly when lies between and , one stretch on each side of .
🧱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?
Taking away from all three parts slides them the same distance along the line, so none of them changes its place in the order.
🧱Adding to both sides keeps an inequalitySliding both numbers the same distance along the number line keeps the one on the right on the right, so adding or subtracting the same amount keeps an inequality's direction.
2STEP 2
Case a ＞ 0
Dividing by a positive a keeps the signs. Matching endpoints and adding the two equations gives a=-1, which contradicts a ＞ 0.
-b-1=-2a, b-1=4a → -2=2a → a=-1
The only candidate breaks its own assumption, so this case is eliminated.
8.EE.C.8Eliminate Possibilities3STEP 3
Case a ＜ 0
Dividing by a negative a flips both signs. Matching endpoints and adding the equations gives a=-1, and then b=1-2a=3.
b-1=-2a, -b-1=4a → a=-1, b=1-2(-1)=3
Here a=-1 ＜ 0 and b=3 ≥ 0, so the case is consistent.
8.EE.C.8Identify Subproblems4STEP 4
Check
Substitute a=-1, b=3 back into the original inequality.
|-x+1| ≤ 3 → -3 ≤ 1-x ≤ 3 → -2 ≤ x ≤ 4
The solution set is exactly the given interval.
7.EE.B.4Convert To AlgebraAnswer
a=-1, b=3
With a=-1, b=3: x=-2 gives |3|=3 ≤ 3 and x=4 gives |-3|=3 ≤ 3, both on the boundary, while x=5 gives |-4|=4 ＞ 3. So the endpoints are right.
Takeaway
When you divide by a letter, ask first whether it is positive or negative.
- Remove the absolute value
- Case a ＞ 0
- Case a ＜ 0
- Check
Where next?
Another one like thissuggested
Step back and solidifysuggested
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