03/14 3. Inequalities Free Online FlashCards

When must an inequality symbol be reversed?

Reverse the inequality symbol when multiplying or dividing both sides by a negative number.

Solve $4x-7\le 9$.

The solution is $x\le 4$, which is $(-\infty,4]$ in interval notation.

What happens when simplifying removes the variable?

A true statement after simplification gives all real numbers; a false statement gives no solutions, written $\varnothing$.

What does “and” mean in a compound inequality?

An “and” compound inequality represents the intersection of its solution sets: values satisfying both conditions.

What does “or” mean in a compound inequality?

An “or” compound inequality represents the union of its solution sets: values satisfying at least one condition.

Solve $|2x-3|=7$.

Set the expression equal to both $7$ and $-7$: $2x-3=7$ or $2x-3=-7$. Thus, $x=5$ or $x=-2$.

Solve $|x-4|\le 3$.

Rewrite it as $-3\le x-4\le 3$, then solve: $1\le x\le 7$, or $[1,7]$.

Solve $|2x+1|>5$.

Rewrite it as $2x+1<-5$ or $2x+1>5$. The solution is $x<-3$ or $x>2$.

What do brackets and parentheses mean in interval notation?

A bracket means the endpoint is included; a parenthesis means the endpoint is excluded.

Why does infinity always have a parenthesis?

Infinity always uses a parenthesis because infinity is not a real number and cannot be included.

How do inequality symbols determine number-line graphs?

Use an open circle for $<$ or $>$, and a closed circle for $\le$ or $\ge$. Shade the region containing the solutions.

How many notebooks can cost at most $\$80$ after a $\$20$ fee?

Model the cost with $20+6n\le 80$. Since $n$ counts notebooks, the meaningful solutions are $n\in\{0,1,2,\ldots,10\}$.