This question has two parts. a) On the number line, show the inequality negative one is less than 𝑥 is less than or equal to four. Part b) Solve the inequality six is greater than or equal to seven 𝑡 minus four.
Beginning with part a, let’s go ahead and take this inequality and break it up. This half says 𝑥 is less than or equal to four. This symbol means less than and this symbol means equal to. So on our number line, here’s four. We can be equal to four and we can be less than four. So numbers less than four will be to the left on our number line for 𝑥.
And to represent the equal to part, the fact that we can be equal to four means we need to have a closed circle above four. Now, for this part, negative one is less than 𝑥. However, when we look at inequalities, we usually have the variable on the left-hand side. However, if we switch these places, we also need to switch the symbol, the sign. So instead of less than, it’s greater than.
Now, a nice way to just double-check is the less than and greater than symbols kind of look like a mouth. And to begin, the mouth was open towards 𝑥, like it was going to take a bite of 𝑥. So when we switch this, we need to make sure the mouth is still facing 𝑥. And it is. So we need to represent 𝑥 is greater than negative one.
So here’s negative one. All of the numbers greater than negative one are to the right of negative one because that’s where the numbers are more positive or become more positive. However, we can’t be equal to negative one. So instead of having a closed circle like we did at four, we need to have an open circle.
So we know what kind of circles we need on top of negative one and four. And now, we need where they overlap. So to make our final picture, we have an open circle at negative one, a closed circle at four, and then everything in between. So this would represent our inequality on a number line.
Now, for part b, we need to solve the inequality six is greater than or equal to seven 𝑡 minus four. Now, we will solve this inequality very similar to how we would solve an equation. So we need to get 𝑡 by itself. So we need to add four to both sides of the inequality. so we have 10 is equal to seven 𝑡 because the four is cancelled on the right and on the left six plus four is equal to 10.
So now, to get 𝑡 by itself, we’ll need to get rid of seven. Seven is being multiplied to 𝑡. And the opposite of multiplying to 𝑡 would be to divide. And when we do so, the sevens cancel on the right. And we have that ten sevenths is greater than or equal to 𝑡.
However, like we said before, the variable is usually on the left-hand side. So in our original one, the greater than or equal to symbol was opening up towards ten sevenths. So we need to make sure and do that here. So we have that 𝑡 is less than or equal to ten sevenths. And these will be our final answers.