Solve the inequality five 𝑚 minus
nine multiplied by 𝑚 plus three is less than 14 in the set of rational numbers.
To solve this inequality in the set
of rational numbers, we need to apply the axioms of inequalities to isolate the
variable. The variable currently appears in
two places, so we’ll begin by distributing the parentheses on the left-hand
side. This gives five 𝑚 minus nine 𝑚
minus 27 is less than 14.
Next, we group the 𝑚-terms
together. Five 𝑚 minus nine 𝑚 is negative
four 𝑚. So we have negative four 𝑚 minus
27 is less than 14. To isolate the 𝑚-term, we then add
27 to both sides of the inequality. This gives the equivalent
inequality negative four 𝑚 is less than 14 plus 27, which simplifies to negative
four 𝑚 is less than 41.
The next step is to divide both
sides of the inequality by negative four, but we must be really careful here. When we multiply or divide both
sides of an inequality by a negative value, this reverses the direction of the
inequality. So on the left-hand side, we have
𝑚. On the right-hand side, we have
negative 41 over four. And we reverse the direction of the
inequality sign to become a greater than sign. We therefore have 𝑚 is greater
than negative 41 over four.
We are asked to solve this
inequality in the set of rational numbers. So we can give our solution in set
notation as the set of all values of 𝑚 such that 𝑚 is a rational number and 𝑚 is
greater than negative 41 over four.