Question

decide whether the statement is true or false. if it is false, explain why. the union of the solution sets of 5x + 1 = 26, 5x + 1 > 26, and 5x + 1 < 26 is ∅. choose the correct answer below. a. false because the union is {5}. b. false because the union is (-∞, 5)∪(5, ∞). c. false because the union is (-∞, ∞). d. true

Explanation:

Step1: Solve \(5x + 1 = 26\)

Subtract 1 from both sides: \(5x = 26 - 1 = 25\). Then divide by 5: \(x = \frac{25}{5} = 5\). So the solution set for the equation is \(\{5\}\).

Step2: Solve \(5x + 1 > 26\)

Subtract 1: \(5x > 25\). Divide by 5: \(x > 5\). Solution set is \((5, \infty)\).

Step3: Solve \(5x + 1 < 26\)

Subtract 1: \(5x < 25\). Divide by 5: \(x < 5\). Solution set is \((-\infty, 5)\).

Step4: Find the union

The union of \(\{5\}\), \((5, \infty)\), and \((-\infty, 5)\) is all real numbers, because \((-\infty, 5)\) includes all numbers less than 5, \(\{5\}\) includes 5, and \((5, \infty)\) includes all numbers greater than 5. So the union is \((-\infty, \infty)\).

Answer:

C. False because the union is \((-\infty, \infty)\).