不等式

$x>3$ means “$x$ is greater than 3” (3 not included), while $x\ge 3$ means “$x$ is greater than or equal to 3” (3 included). The same applies to $<$ and $\le$. Whether the boundary value itself is included is exactly the distinction between the strict symbols $<,>$ and the inclusive symbols $\le ,\ge$.

The basics are the same as for equations: add or subtract the same quantity on both sides, or multiply/divide by a positive number, to isolate $x$. Example: from $3x+1<16$, subtract 1 to get $3x<15$, then divide by $3$ (positive) to get $x<5$. As long as you divide by a positive number, the sign stays the same.

This is the single most common mistake with inequalities. Whenever you multiply or divide by a negative number, you must reverse the direction of the sign. Example: $5-2x>1$. Subtracting 5 gives $-2x>-4$; dividing both sides by $-2$ reverses $>$ into $<$, giving $x<2$. To avoid the flip altogether, you can instead move the $x$-term to the side where its coefficient is positive.

Represent the solution on a number line. Use an open circle (○) when the boundary is not included ($<$ or $>$) and a filled circle (●) when it is included ($\le$ or $\ge$), then draw an arrow in the direction the solution extends. For $x\ge -3$, place a filled circle at $-3$ and draw the arrow to the right.

A double inequality such as $-7\le 2x-1<9$ has an expression in $x$ trapped between two values. The trick is to apply the same operation to all three parts. Add 1 to each part to get $-6\le 2x<10$, then divide each part by 2 to get $-3\le x<5$. If you divide by a negative, reverse both signs at once.

When asked to “list all integers that satisfy” a condition, first solve the inequality to get a range, then write out the integers inside it, watching the endpoints. Example: the integers satisfying $-3\le x<5$ are $-3,-2,-1,0,1,2,3,4$. Because of $<5$, 5 is excluded; because of $-3\le$, $-3$ is included — getting these endpoints right is where the marks are won or lost.