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14. Reason Abstractly Compare and contrast the solutions \( x<2 \) and \( x \leq 2 \).

Ask by Spencer Barber. in the United States
Mar 11,2025

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\( x < 2 \) includes all numbers less than 2, excluding 2, while \( x \leq 2 \) includes all numbers less than or equal to 2, including 2.

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Bonus Knowledge

When we think of \( x < 2 \), we're talking about all the numbers that are less than 2, which means we can have 1.9, 1, or even fractions like 1.5—basically anything that tiptoes below the magical barrier of 2, but never touching it! Meanwhile, \( x \leq 2 \) includes everything less than 2 as well, but with a special VIP pass for the number 2 itself, since it's now included in our party! So, while both sets share all values below 2, only \( x \leq 2 \) allows us to have a gab fest with the number 2. Visualizing these solutions on a number line can be super helpful! For \( x < 2 \), you'd draw an open circle at 2 to show it's not included, and shade everything to the left. For \( x \leq 2 \), you'd use a closed circle at 2 to show that it’s included in our club. These visual aids can help clarify just how these two inequalities differ, even if they seem quite similar on the surface!

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