 Please look at the "never true," "sometimes true" and "always true" table that shows the algebraic expressions that fit below each category. I just need someone to explain to me how they are the same and different (please don't answer the other questions)

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The picture you have provided is too indistinct to make out the table.

However, to help you I have given examples of always, sometimes, never true.

Always true include identities:

Your birthday falls on a day this year.

3x+2x=5x, sin²(x)+cos²(x)=1, 7×4=2×14, sin(x)=cos(π/2-x), -1≤sin(x)≤1.

Sometimes true:

x²-x-2=0 is true for x=2 and x=-1 only because x²-x-2=(x-2)(x+1)=0 which is true when x-2=0, that is, x=2; or x+1=0, that is, x=-1.

The sky is blue.

sin(x)=½ is true for many values of x (x=π/6, x=5π/6, ...).

2x+3y=10 is true for many values of x and y, for example, x=5 and y=0; x=0, y=3⅓, x=½, y=3.

Never true:

2x+3x>5x, sin(x)>1, 7³=340, -1<sec(x)<1, x-1=x+1, eˣ=-2.

The sun is visible at night.

Sometimes true implies that the equation or inequality can be solved, that is, specific value(s) of a variable(s) can be found that satisfy the equality or inequality, and there may be lots of solutions (maybe an infinite number). However, the equation or inequality is not true for every possible value of the variable(s). But never true statements have no solutions. Always true statements are true for all values of the variables involved.

A statement may be sometimes, always or never true at first sight. It’s only when you try to solve an equation or inequality that you discover whether or not it can be solved, because the appearance of a statement isn’t a guide to whether it’s true, or false, or sometimes true. In this way the three types are similar. An equation may be true for some values of its variables, but it may also be an identity (always true) which means there are no specific values to “solve” it, because all values make it true. This is what makes the difference between the three types of statement. You prove an identity, rather than solve it—that’s the difference between sometimes true and always true.

by Top Rated User (721k points)