Question Video: Solving Multistep Linear Inequalities over the Set of Rational Numbers Mathematics • First Year of Preparatory School

Video Transcript

Solve the inequality seven 𝑥 minus eight 𝑥 plus 11 is less than or equal to eight in the set of rational numbers.

To solve this inequality, we need to apply the axioms of inequalities to isolate the variable. We begin by combining the 𝑥-terms on the left-hand side. Seven 𝑥 minus eight 𝑥 is negative 𝑥, so we have negative 𝑥 plus 11 is less than or equal to eight. Next, we isolate the variable by subtracting 11 from each side to give the equivalent inequality negative 𝑥 is less than or equal to negative three.

The next step is either to multiply or divide both sides of the inequality by negative one so that the coefficient of 𝑥 is simply one. But we need to be really careful here. One of the key axioms of inequalities is that when we multiply or divide both sides of an inequality by a negative number, this reverses the direction of the inequality. So negative 𝑥 multiplied by negative one is 𝑥. Negative three multiplied by negative one is three. And we reverse the inequality sign from less than or equal to to greater than or equal to, giving 𝑥 is greater than or equal to three.

We’re asked to solve this inequality only in the rational numbers. So we can express our solution as the set of all values of 𝑥 such that 𝑥 is a rational number and 𝑥 is greater than or equal to three.