why cross-multiplication is NOT the right method to use in solving the following inequality
(x-2)/(x+5) ≤ 3/5
Cross-multiplication is a method typically used to solve equations, not inequalities. When solving an inequality, it is important to preserve the direction of the inequality sign. In the case of the inequality (x-2)/(x+5) ≤ 3/5, cross-multiplication would not be the right approach.
To solve this inequality, we need to first get rid of the fractions by multiplying both sides of the inequality by the denominators. In this case, the denominators are (x+5) on the left side and 5 on the right side.
(x+5) * (x-2)/(x+5) ≤ (x+5) * 3/5
Simplifying:
x - 2 ≤ 3(x + 5) / 5
Now, we can simplify the right side of the equation:
x - 2 ≤ (3x + 15) / 5
To isolate the variable x, we can multiply both sides of the inequality by 5 to get rid of the fraction:
5(x - 2) ≤ 3x + 15
Expanding:
5x - 10 ≤ 3x + 15
Now, we can move the terms involving x to one side of the equation and the constant terms to the other side:
5x - 3x ≤ 15 + 10
2x ≤ 25
Finally, we can divide both sides by 2 to solve for x:
x ≤ 12.5
Therefore, the solution to the inequality is x ≤ 12.5.
As you can see, the process of solving this inequality involved multiplying both sides of the inequality by the denominators, simplifying, and then isolating the variable. Cross-multiplication, on the other hand, would not have been an appropriate method for solving this inequality.