document.write( "Question 1084270: It is tempting to try to solve an inequality like an equation. For instance, we might try to solve 1<3/x
\n" ); document.write( "by multiplying both sides by x, to get x < 3, so the solution would be (-∞ ,3)
\n" ); document.write( "But that’s wrong; for example, x = -1 lies in this interval but does not satisfy the original inequality. Explain why this method does not work. In addition, solve the inequality correctly. Please be sure to explain your reasoning! \n" ); document.write( "

Algebra.Com's Answer #698350 by KMST(5328) $\"\"$ $\"About$

You can put this solution on YOUR website!
With an equation, multiplying both sides by a number other than zero gives you an equivalent equation, with exactly the same solutions. Multiplying times zero gives you 0=0, which in most cases would have more solutions than the original equation. For equations, if we multiply times an expression involving x, we could be gaining solutions that do not work for the original equation (extraneous solutions).
\n" ); document.write( "
\n" ); document.write( "For inequalities, it gets more complicates, not only we have to worry about factors that could be zero. we also have to worry about if a factor is negative or positive.
\n" ); document.write( "For an inequality, multiplying times a non-zero factor both sides of the inequality sign, and keeping the inequality sign as is, will give you an equivalent inequality only if the factor is positive.
\n" ); document.write( "For example 1<2 , and multiplying times 3, we get 3<6, which is equally true.
\n" ); document.write( "However, if we multiply times -5 both sides, we would get -5 and -10,
\n" ); document.write( "and we know that -5 > -10.
\n" ); document.write( "For an inequality, multiplying times a negative number both sides of the inequality sign, and turning around the inequality sign, will give you an equivalent inequality.
\n" ); document.write( "With $\"1%3C3%2Fx\"$ we know that $\"x=0\"$ is not a solution.
\n" ); document.write( "We could multiply times $\"x\"$ and solve $\"x%3C3\"$ , but that means we are assuming $\"x%3E0\"$ .
\n" ); document.write( "Could it be that $\"x%3E0\"$ ? Assuming $\"x%3E0\"$ , we could multiply both sides times x, and get the equivalent equation $\"x%3C3\"$ , which does not quite contradict the assumption that $\"x%3E0\"$ . So, we know that x values such that $\"0%3Cx%3C3\"$ are solutions of $\"1%3C3%2Fx\"$ .\r
\n" ); document.write( "\n" ); document.write( "Could it be that there are any solutions with $\"x%3C0\"$ ? Assuming $\"x%3C0\"$ , an equivalent inequality would be $\"x%3E3\"$ , which is not true for any $\"x%3C0\"$ . So, there are no other solutions. \n" ); document.write( "

\n" );