Video Transcript
Which of the following number lines represents the solution set of the inequality seven multiplied by 𝑥 minus two minus eight is less than five 𝑥 minus 10 in the set of natural numbers?
Let’s work through the process of solving this inequality. And then we can consider how to represent its solution on a number line.
We’ll begin by distributing the seven over the parentheses on the left-hand side to give seven 𝑥 minus 14 minus eight is less than five 𝑥 minus 10. Next, we can group the constants on the left-hand side together to give seven 𝑥 minus 22 is less than five 𝑥 minus 10. We now have 𝑥 terms on both sides of the inequality, as well as constant terms on both sides.
To collect the 𝑥-terms on the same side first, let’s subtract five 𝑥 from both sides. That gives two 𝑥 minus 22 is less than negative 10. Next, we can collect the constants on the other side of the inequality by adding 22 to both sides to give two 𝑥 is less than 12. Finally, we divide both sides of the inequality by two to give 𝑥 is less than six.
Remember that when we divide or multiply both sides of an inequality by a negative value, we have to reverse the direction of the inequality. But this doesn’t apply here as the value we’re dividing by is positive.
We’ve now solved the given inequality. And we need to determine which of the five given options represents this solution on a number line. Before we do, however, there is some other important information given in the question. We’re asked which number line represents the solution set of the given inequality in the set of natural numbers.
The natural numbers can be thought of as those numbers that are used for counting, together with zero and hence include all the nonnegative integers. So, thinking about all the nonnegative integers that are strictly less than six, we find that the solution to this inequality in the natural numbers is the set containing the values zero, one, two, three, four, and five. Note that six is excluded because the inequality is strict.
On a number line, this set would be represented by individual dots placed above each value. We can see that this is represented in option (B). There are dots above each of the integers from zero to five inclusive. So the answer is (B). This number line represents the solution to the given inequality in the set of natural numbers.
