ecall the definition of the absolute value function: for x ∈ R, |x| = xif x ≥ 0 and |x| = −x if x ≤ 0. Using the definition of absolute valueand that |x| = | − x| for all x ∈ R, do a proof by cases to prove thatfor all a and b in R that |ab| = |a||b|

Sure, let's prove that |ab| = |a||b| for all a and b in R using the definition of absolute value and proof by cases.

Case 1: a ≥ 0 and b ≥ 0 In this case, |a| = a and |b| = b. Therefore, |ab| = ab = a*b = |a||b|.

Case 2: a ≥ 0 and b ≤ 0 In this case, |a| = a and |b| = -b. Therefore, |ab| = a*(-b) = -ab = |a||b|.

Case 3: a ≤ 0 and b ≥ 0 In this case, |a| = -a and |b| = b. Therefore, |ab| = (-a)*b = -ab = |a||b|.

Case 4: a ≤ 0 and b ≤ 0 In this case, |a| = -a and |b| = -b. Therefore, |ab| = (-a)*(-b) = ab = |a||b|.

In all cases, we have shown that |ab| = |a||b|. Therefore, we have proven that for all a and b in R, |ab| = |a||b|.