Question Video: Representing the Solution Set of an Inequality on a Number Line Mathematics • 9th Grade

Which of the following diagrams represents the inequality (√2)𝑥 > √8? [A] Diagram A [B] Diagram B [C] Diagram C [D] Diagram D [E] Diagram E

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Video Transcript

Which of the following diagrams represents the inequality square root of two times 𝑥 is greater than the square root of eight?

The first diagram shows an open circle at two and includes all the real numbers to the right. This means 𝑥 is greater than two. The next diagram shows an open circle at two and includes all the real numbers to the left. This means 𝑥 is less than two. The third diagram shows a shaded circle on two and all the real numbers to the right. This means 𝑥 is greater than or equal to two. The fourth diagram shows an open circle at the square root of two and all real numbers to the left. This means 𝑥 is less than the square root of two. The final diagram shows an open circle at the square root of two and all real numbers to the right. This means 𝑥 is greater than the square root of two.

In order to solve the given inequality, we need to isolate 𝑥 on the left side and match our solution to one of these five diagrams. We can do this by dividing the inequality through by the square root of two. This leads to the inequality 𝑥 is greater than the square root of eight divided by the square root of two. We note that the inequality sign changes direction whenever we multiply or divide by a negative number. Since this inequality involves only positive real numbers, we leave the sign as we found it.

We now recall the quotient property for square roots, which says that the square root of 𝑎 divided by the square root of 𝑏 is equal to the square root of 𝑎 divided by 𝑏, for 𝑎 greater than or equal to zero and 𝑏 greater than zero. This means we can write the right-hand side of the inequality as the square root of eight divided by two, which simplifies to the square root of four. And we know that the inequality 𝑥 greater than the square root of four is equivalent to 𝑥 greater than two.

Finally, we find that our answer matches the inequality represented by diagram (A). This diagram shows the solution set of all real numbers greater than two, which satisfies the given inequality.

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