Question Video: Solving Multistep Linear Inequalities over the Set of Rational Numbers | Nagwa Question Video: Solving Multistep Linear Inequalities over the Set of Rational Numbers | Nagwa

Question Video: Solving Multistep Linear Inequalities over the Set of Rational Numbers Mathematics • 7th Grade

Solve the inequality 5𝑚 − 9(𝑚 + 3) < 14 in ℚ.

02:09

Video Transcript

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.

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