Question

How do you write the interval \[[3,4]\] as an inequality involving x and show each inequality using the real number line?

Answer 1

Hint: An inequality that includes the boundary point indicated by the “or equal” part of the symbols \[\underline{<}\] and \[\underline{>}\] and a closed dot on the number line. The symbol \[(\infty )\]indicates the interval is unbounded to the right. Express ordering relationships using the symbol \[<\] for “less than” and \[>\] for “greater than.”

Complete step by step answer:
An algebraic inequality, such as \[x\underline{>}3\], is read “x is greater than or equal to 3”. This inequality has infinitely many solutions for x. Some of the solutions are \[3,5,6.5,19\]and \[36.009\]. Since it is impossible to list all of the solutions, a system is needed that allows a clear communication of this infinite set. Two common ways of expressing solutions to inequality are by graphing them on a number line and using interval notation.
Square brackets denote that the values listed in the interval are included in the interval. Whereas round brackets will not include the values listed in the interval.
So, we can write the interval \[[3,4]\] as the inequality \[3\underline{<}x\underline{<}4\].
The below figure shows the \[[3,4]\] interval on a real number line.

Note:
While solving such types of problems, there is a scope of making mistakes like keeping a closed dot for the unbounded value of the interval. Check whether the round bracket is present or the square bracket is present.