SAT Math 4: Linear Inequalities

Inequality

A mathematical relation comparing two quantities that are not necessarily equal. Common inequality symbols include $<$, $>$, $\leq$, and $\geq$.

How do strict and non-strict inequalities differ?

Strict inequalities, such as $a<b$ or $a>b$, exclude equality. Non-strict inequalities, such as $a\leq b$ or $a\geq b$, allow the two sides to be equal.

What do the symbols $a<b$, $a>b$, $a\leq b$, and $a\geq b$ mean?

$a<b$ means $a$ is less than $b$; $a>b$ means $a$ is greater than $b$; $a\leq b$ means $a$ is less than or equal to $b$; and $a\geq b$ means $a$ is greater than or equal to $b$.

How are the inequalities $a\leq b$ and $b\geq a$ related?

They are equivalent statements; each is the converse of the other.

Transitive property of inequalities

If $a\leq b$ and $b\leq c$, then $a\leq c$. If at least one of the two original inequalities is strict, the conclusion is strict.

What happens when the same number is added to or subtracted from both sides of an inequality?

The direction of the inequality does not change. For example, if $a\leq b$, then $a+c\leq b+c$ and $a-c\leq b-c$.

What happens when both sides of an inequality are multiplied or divided by a positive number?

The inequality direction is preserved: if $a\leq b$ and $c>0$, then $ac\leq bc$ and $a/c\leq b/c$.

Why must an inequality sign be reversed when both sides are multiplied or divided by a negative number?

Multiplication by a negative number reverses the order of numbers on the number line. Thus, if $a\leq b$ and $c<0$, then $ac\geq bc$ and $a/c\geq b/c$.

How does taking the additive inverse affect an inequality?

It reverses the direction. If $a\leq b$, then $-a\geq -b$.

How does the sign of a function's monotonicity determine whether an inequality reverses?

A monotonically increasing function preserves the inequality direction, while a monotonically decreasing function reverses it. Strictness is preserved when the function is strictly monotonic and the original inequality is strict.

What is the correct interpretation of chained notation $a<b<c$?

It means both $a<b$ and $b<c$. By transitivity, it also implies $a<c$.

How should a chained inequality such as $4x<2x+1\leq3x+2$ be solved?

Solve each adjacent inequality separately: $4x<2x+1$ gives $x<1/2$, while $2x+1\leq3x+2$ gives $x\geq-1$. Their intersection is $-1\leq x<1/2$.

What is a system of inequalities?

A system is a collection of inequalities that must all be satisfied simultaneously. Its solution is the intersection of the individual solution sets, often represented as a feasible region when variables are graphed.

What is a feasible region?

A feasible region is the set of points that satisfy all inequalities in a system simultaneously. For systems involving multiple variables, it can be represented as a region in the coordinate plane.