Evaluate the following expression:

Concept:

When an expression contains absolute-value bars, treat each pair as a grouping symbol: first evaluate what is inside, then take the non-negative (absolute) value of that result. Then apply the standard order of operations (BODMAS) — brackets/absolute values first, then multiplication and division from left to right, then addition and subtraction from left to right. When the expression is a fraction, evaluate the numerator and the denominator independently using this rule, and divide only at the end.

Application:

1. Evaluate the absolute values: |3 − 5| = |−2| = 2 and |4| = 4.

2. Substitute into the numerator: 9|3 − 5| − 5|4| ÷ 10 = 9(2) − 5(4) ÷ 10.

3. Apply multiplication and division left to right in the numerator: 5(4) ÷ 10 = 20 ÷ 10 = 2, so the numerator = 9(2) − 2 = 18 − 2 = 16.

4. Substitute into the denominator: −3(5) − 2 × 4 ÷ 2.

5. Apply multiplication and division left to right in the denominator: −3(5) = −15 and 2 × 4 ÷ 2 = 8 ÷ 2 = 4, so the denominator = −15 − 4 = −19.

6. Divide the numerator by the denominator: 16 ÷ (−19) = −16/19.

Cross-check:

Both tied multiplication/division chains here are unambiguous: 5 × 4 ÷ 10 gives 2 whether computed as (5 × 4) ÷ 10 or as 5 × (4 ÷ 10), and 2 × 4 ÷ 2 gives 4 either way — so the left-to-right rule and any other grouping of the tied operators agree, confirming the numerator and denominator values are not order-dependent.

So the expression evaluates to 16/(−19), i.e. −16/19.