Question

determine how many solutions exist for the given equation. 3(-7x - 5) - 1 = -21x - 11 the equation is an identity with infinitely many solutions; the solution set is all real numbers. the equation is a contradiction; the solution set is the empty set.

Explanation:

Step1: Expand the left - hand side

Using the distributive property \(a(b + c)=ab+ac\), for \(3(-7x - 5)-1\), we have:
\(3\times(-7x)+3\times(-5)-1=-21x-15 - 1\)
Simplify the constants: \(-21x-16\)

Step2: Compare both sides of the equation

The original equation is \(3(-7x - 5)-1=-21x - 11\). After expanding the left - hand side, we get \(-21x-16=-21x - 11\)
Add \(21x\) to both sides of the equation:
\(-21x+21x-16=-21x + 21x-11\)
Simplify both sides: \(-16=-11\)

Since \(-16
eq - 11\), this is a contradiction. A contradiction equation has no solutions, so the solution set is the empty set.

Answer:

The equation is a contradiction; the solution set is the empty set.