### Video Transcript

Circle the inequality shown in the diagram. Is it negative three is less than or equal to π₯ is less than or equal to seven, negative three is less than or equal to π₯ which is less than seven, negative three is less than π₯ which is less than seven, or negative three is less than π₯ which is less than or equal to seven?

Notice that all four of the options that weβve been given have the same value at the lower end of the interval. Itβs negative three. They also all have the same value of seven at the upper end of the interval. This makes sense when we look at the diagram. The line extends all the way from negative three to seven, meaning that this inequality represents π₯-values between negative three and seven.

What differs between the four options are the inequality signs that have been used. Some are strict inequalities β π₯ is less than seven β whereas others are weak inequalities β π₯ is less than or equal to seven. We need to decide what the correct pair of inequality signs is.

Notice that we have a solid or closed circle at the lower end of the interval. But we have an open circle at the upper end. We need to remember what these pieces of notation mean.

A closed circle represents a weak inequality, greater than or equal to or less than or equal to. This means the value at the closed circle is included in the interval. So we need a weak inequality sign at the lower end of the interval so that negative three is included as a possible answer. There were only two possibilities which have the correct sign here, the first two.

An open circle, however, represents a strict inequality, strictly greater than or strictly less than, meaning that the value here is not included in the interval. So we need a strict inequality sign at the upper end of the interval. Again, there were two possibilities which have the correct sign. But this time, theyβre the second and third options.

The only one of the four options which has the correct signs in both places is the second one: negative three is less than or equal to π₯ is less than seven. This means that π₯ can take any value from negative three to seven, including negative three but excluding seven.