Additional Mathematics Synthesis Guide 46: Case Splitting and Branch Control — Equations, Inequalities, Trigonometric Intervals and Domain Conditions

Detect the branch point → write each case explicitly → solve each case under its own conditions → intersect with the original domain → check completeness.

1. What creates a branch?

A branch appears whenever the mathematics allows more than one logically distinct possibility.

• A product equals zero.
• A square equals a positive quantity.
• An inequality changes sign at critical values.
• A trigonometric value occurs at several angles in an interval.
• A parameter can lie in different regimes.
• A transformation is only valid under a condition such as a denominator being non-zero.

2. Product-zero structure

If

(x−2)(x+5)=0,

then either x−2=0 or x+5=0. Hence x=2 or x=−5.

The word “or” matters. The factorisation creates two branches, and both must be considered.

3. Division can erase a branch

From

x(x−3)=0,

dividing by x gives x−3=0 and loses x=0.

Division by a variable expression silently assumes that expression is non-zero. If zero is possible, split into the zero branch and the non-zero branch instead.

4. Square equations create sign branches

If

(x−4)²=9,

then

x−4=3 or x−4=−3.

Thus x=7 or1.

Taking only the positive square root produces an incomplete solution set.

5. Square-root equations need sign conditions before squaring

For

√(x+1)=x−1,

the right side must be non-negative, so x≥1. Squaring then gives candidates x=0 and3, but only x=3 lies in the valid branch and satisfies the original equation.

Squaring merges positive and negative possibilities, so the original sign branch must be restored afterward.

6. Denominator conditions define legal branches

For

(x+1)/(x−2)=3,

the original domain requires x≠2. Multiplying by x−2 is valid only inside that branch.

After solving, x=7/2 is admissible; x=2 would never have been a candidate because the original expression is undefined there.

7. Inequalities are branch problems

Consider

(x−1)(x−4)>0.

The critical values x=1 and4 split the number line into three branches:

• x<1;
• 1<x<4;
• x>4.

The product is positive when the factors have the same sign, giving x<1 or x>4.

8. Endpoint inclusion is part of the branch

If the inequality were

(x−1)(x−4)≥0,

the endpoints x=1 and4 would be included. A strict inequality and a non-strict inequality therefore have different branch boundaries.

9. Rational inequalities need denominator branches

For

(x−3)/(x+1)>0,

critical values are x=3 from the numerator and x=−1 from the denominator. The point x=−1 is excluded regardless of the inequality symbol because the expression is undefined there.

Testing intervals gives x<−1 or x>3.

10. Sign charts are controlled case tables

A sign chart is not merely a drawing. It is a compact table of cases separated by critical values. Each row or interval records the sign of each factor and therefore the sign of the full expression.

This is especially useful when several factors create many possible sign combinations.

11. Trigonometric equations generate angle branches

For

sinx=1/2, 0≤x≤2π,

the principal reference angle is π/6, but sine is positive in two positions over one full cycle. The complete interval solutions are

x=π/6, 5π/6.

One inverse-trig output is not automatically the full answer.

12. Factorised trigonometric equations preserve branches

For

sinx(2cosx−1)=0,

solve both branches:

• sinx=0;
• 2cosx−1=0.

Dividing by sinx would destroy the first branch.

13. Interval restrictions filter periodic branches

Trigonometric equations have repeating solutions over all real numbers, but school questions usually ask for solutions in a specified interval. The interval acts as a branch filter.

Generate all candidates relevant to the period structure, then keep only those lying inside the requested interval.

14. Principal inverse values are starting points, not complete sets

A calculator may return one principal value for sin⁻¹, cos⁻¹ or tan⁻¹. The graph symmetry and period of the original trig function determine whether additional interval solutions exist.

Guide 28 develops this interval-completion process in depth.

15. Parameter questions create regime branches

For the family

x²−4x+k=0,

the discriminant 16−4k divides parameter space into three branches:

• k<4: two distinct real roots;
• k=4: one repeated root;
• k>4: no real roots.

The threshold value is itself a separate case.

16. “Always positive” and “non-negative” are different branches

For an upward-opening quadratic, strict positivity for all real x requires the graph to stay above the x-axis, usually corresponding to Δ<0. Non-negativity allows tangent contact at zero, so Δ≤0.

The equality case must be treated separately because the wording changes whether the threshold is included.

17. Domain intersections combine branch conditions

For

ln(x−1)/√(5−x),

we need x>1 from the logarithm and x<5 from the square root in the denominator. The valid branch is the intersection:

1<x<5.

18. Composite functions create nested branches

For f(x)=lnx and g(x)=x²−4, f(g(x)) exists only when g(x)>0. Thus x²−4>0, giving

x<−2 or x>2.

The inner expression creates two valid outer-domain branches.

19. Connected rates may need geometric branches

If a geometric relationship contains a square such as x²+y²=r², solving for y gives ±√(r²−x²) algebraically. A physical diagram may restrict the situation to the upper half, lower half or positive lengths only.

The diagram and variable definitions decide which algebraic branch represents the model.

20. Kinematics uses sign branches

Velocity zeros divide time into motion-direction intervals. To compute total distance, one must split at each actual direction change and add distance magnitudes across the branches.

Integrating velocity over the whole interval gives displacement, not total distance, because positive and negative branches cancel.

21. Completeness is a proof obligation

A solution is complete only when every logically possible branch has either been solved or excluded for a stated reason.

Useful completeness questions include:

• Did I divide by something that could be zero?
• Did a square create ± possibilities?
• Did an inverse trig function return only a principal value?
• Did I test every sign interval?
• Did I include or exclude threshold endpoints correctly?
• Did I apply all original domain conditions?

22. A reliable branch-control routine

1. Scan the original problem for branch points: products, squares, denominators, trig functions, inequalities, parameters and domain restrictions.
2. Record all legal-domain conditions before transforming.
3. When a zero factor is possible, split instead of dividing it away.
4. When a sign choice appears, write each case explicitly.
5. For inequalities, mark critical values and test every interval.
6. For trig equations, generate every solution in the stated interval, not only the principal one.
7. Apply endpoint rules and domain exclusions.
8. Substitute or structurally verify surviving candidates.
9. State the final solution set in a form that makes the branch union clear.

23. Common failure patterns

• Dividing by a factor that may be zero and losing a solution branch.
• Taking only the positive root of a square equation.
• Squaring an equation and accepting every resulting root.
• Ignoring denominator exclusions in rational equations or inequalities.
• Testing only one interval in a sign-chart problem.
• Using one inverse-trig output as the entire interval solution.
• Forgetting a reflected or periodic trig solution.
• Using Δ≤0 when the wording requires strict positivity.
• Combining domain conditions by union when they should be intersected.
• Giving several candidates without explaining why the final set is complete.

24. Practice set

1. Solve (x−2)(x+5)=0.
2. Explain why dividing x(x−3)=0 by x is unsafe.
3. Solve (x−4)²=16.
4. For √(x+1)=x−1, state the sign condition before squaring.
5. What domain applies to 1/(x−2)?
6. Solve (x−1)(x−4)>0.
7. Solve (x−1)(x−4)≥0.
8. Why is x=−1 excluded from (x−3)/(x+1)>0?
9. Solve sinx=1/2 for 0≤x≤2π.
10. Why should sinx(2cosx−1)=0 not be divided immediately by sinx?
11. What does a calculator inverse-sine value represent?
12. For x²−4x+k=0, what k gives one repeated real root?
13. For an upward quadratic, which discriminant condition supports strict positivity for all real x?
14. Which condition supports non-negativity?
15. Find the domain of ln(x−1)/√(5−x).
16. Find the domain of ln(x²−4).
17. Why can velocity zeros matter for total distance?
18. What does completeness require in a branched solution?
19. When should two domain restrictions be intersected?
20. Name one common branch-loss operation.

Answers

1. x=2 or−5.
2. Because x=0 is a valid branch and division assumes x≠0.
3. x=8 or0.
4. x−1≥0, so x≥1.
5. x≠2.
6. x<1 or x>4.
7. x≤1 or x≥4.
8. The denominator is zero there, so the original expression is undefined.
9. x=π/6,5π/6.
10. The branch sinx=0 would be lost.
11. A principal value; additional interval solutions may exist by symmetry and periodicity.
12. k=4.
13. Δ<0, assuming the leading coefficient is positive.
14. Δ≤0, assuming the leading coefficient is positive.
15. 1<x<5.
16. x<−2 or x>2.
17. They mark possible direction changes, where total-distance calculation must split into branches.
18. Every possible branch must be solved or excluded with a valid reason.
19. When both restrictions must hold simultaneously.
20. Dividing by a variable expression that may equal zero.

25. What mastery looks like

Mastery means the learner can detect branch points before they become errors, preserve zero and sign cases, solve inequalities interval by interval, complete trigonometric solution sets and intersect all domain conditions correctly.

The transfer test is to give a mixed equation with factors, restrictions and an interval. If the learner can explain why the final answer is both valid and complete—rather than merely listing candidates—branch control has become part of mathematical reasoning.

---

Continue through Batch 12

• Guide 45: Symmetry and Invariants — Reflections, Periodicity, Paired Roots and Graph Structure
• Guide 47: Information Filtering and Sufficiency — Relevant Data, Hidden Constraints, Redundancy and Unique Determination
• Guide 48: Comparative Solution Design — Alternative Methods, Reliability, Efficiency and Independent Verification
• Additional Mathematics Directory
• BTT Mathematics Hub