You haven’t submitted a question, so I guess you want some general ideas.

Start with a number line. Let’s mark a point on the line, say, at 5.

The line to the left of 5, but not including 5 itself, is all the numbers less than 5, and we can represent these by writing x<5. When we include 5 itself, we get x≤5, all numbers less than or equal to 5.

The line to the right of 5 is represented by x>5 or x≥5, depending on whether 5 is excluded or included.

I guess you already understand this, so let’s move on.

What does y<x+5 mean? To represent this we can’t use just one number line, but we use two at right angles, one for x (horizontal) and one for y (vertical). Where they cross is the origin (0,0) where both number lines have their zero.

The graph of y<x+5 is a straight line which crosses the x number line (x axis) at -5 and crosses the y number line (y axis) at 5. The line slopes to the right at 45°. The whole area below the line represents y<x+5, but excludes points on the line itself, whereas y≤x+5 includes points on the line. So when we switched from a single number line to two number lines, we went from left and right to lower than (or below) to higher than (or above). Remember L=left=lower=less. y>x+5 and y≥x+5 represent combinations of x and y above the line, excluding or including points on the line itself.

Similarly, we use the same reasoning and representation for, say, y<3x+4, but the line is steeper. Above and below still represent the same things. And no matter what equation we have to relate x and y the same rules apply using above and below (higher and lower).

To solve inequalities we apply the same rules as for solving an equation but take care to preserve the inequality sign. If we have y<2x+7 we can express x in terms of y:

y<2x+7; subtract 7 from both sides: y-7<2x. Now divide through by 2: ½(y-7)<x. Note that the “wide” part of the inequality sign faces x, so if we switch the left and right sides of the inequality we also switch the sign itself: x>½(y-7), and note that the wide part of the sign still faces x.

I hope this helps you.

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