Question

Is the inequality 5(x - 2) ≥ 2x - 1 always true, sometimes true, or never true?

a. Always true
b. Sometimes true
c. Never true

Answer 1

The inequality 5(x - 2) ≥ 2x - 1 is sometimes true, specifically for all x values that are greater than or equal to 3. If x is less than 3, the inequality will not hold.

Step-by-step explanation:

To determine if the inequality 5(x - 2) ≥ 2x - 1 is always true, sometimes true, or never true, we can solve the inequality step by step. First, distribute the 5 to both terms inside the parentheses:

• 
  
• 5(x - 2) = 5x - 10

Now our inequality looks like this:

• 
  
• 5x - 10 ≥ 2x - 1

Next, we want to get all the x terms on one side and constants on the other side, so we subtract 2x from both sides:

• 
  
• 5x - 10 - 2x ≥ 2x - 2x - 1
• 
  
• 3x - 10 ≥ -1

Then, we add 10 to both sides to isolate the term with x:

• 
  
• 3x - 10 + 10 ≥ -1 + 10
• 
  
• 3x ≥ 9

Finally, divide both sides by 3 to solve for x:

• 
  
• 3x / 3 ≥ 9 / 3
• 
  
• x ≥ 3

Therefore, the inequality is sometimes true, specifically for all x values that are greater than or equal to 3. If x is less than 3, the inequality will not hold. Hence, the correct answer is (b) sometimes true.