Question

making an argument your friend says that the inequality ( 5x - 2 > 5x - 4 ) has no solution because the equation ( 5x - 2 = 5x - 4 ) has no solution. is your friend correct: no explain.

Explanation:

Step1: Simplify the inequality

Start with the inequality \(5x - 2>5x - 4\). Subtract \(5x\) from both sides.
\(5x - 2-5x>5x - 4 - 5x\)
Simplifying both sides, we get \(- 2>-4\).

Step2: Analyze the result

The inequality \(-2 > - 4\) is a true statement. It does not depend on the value of \(x\), which means the inequality \(5x - 2>5x - 4\) is true for all real numbers \(x\). While the equation \(5x-2 = 5x - 4\) simplifies to \(-2=-4\) (which is false, so no solution), the inequality has a solution (all real numbers). So the friend is incorrect because the solution of an inequality and an equation are analyzed differently, and the inequality here is always true.

Answer:

The friend is not correct. When we simplify the inequality \(5x - 2>5x - 4\) by subtracting \(5x\) from both sides, we get \(-2>-4\), which is a true statement. This means the inequality is true for all real numbers \(x\) (it has infinitely many solutions), even though the equation \(5x - 2=5x - 4\) has no solution (since it simplifies to \(-2 = - 4\), which is false). The solution of an inequality and an equation are determined in different ways, so the fact that the equation has no solution does not mean the inequality has no solution.