Solving Inequalities Flashcards

All 18 cards in this deck

What does the symbol $\le$ mean?

Less than or equal to.

What is a double inequality?

e.g. $-3 < t \le 2$

An inequality giving both a lower and an upper bound for the unknown.

e.g. $t$ is greater than $-3$ and less than or equal to $2$.

For the inequality $-2 \le n < 5$ where $n$ is an integer, what is the greatest possible value of $n$?

$4$ — because $5$ itself is not included by $<$.

True or false? $x < 5$ and $x \le 5$ give the same list of integer values.

False. $x \le 5$ includes $5$; $x < 5$ does not.

True or false? "No more than 30 people" is written as $p < 30$.

False. It is $p \le 30$, since 30 people is allowed.

On a number line, what does an open (unshaded) circle show?

That the end value is not included, i.e. the sign is $<$ or $>$.

On a number line, what does a closed (shaded) circle show?

That the end value is included, i.e. the sign is $\le$ or $\ge$.

What are the steps to show an inequality with one bound on a number line?

e.g. show $x < 4$

1. Draw the circle at the end value: circle at $4$.
2. Shade it only if the sign is $\le$ or $\ge$: leave it open.
3. Draw a line with an arrow in the direction of the solutions: arrow pointing left.

What are the steps to show a double inequality on a number line?

e.g. show $-3 \le y < 4$

1. Put a circle at each end value: circles at $-3$ and $4$.
2. Shade each circle only if that sign is $\le$ or $\ge$: closed at $-3$, open at $4$.
3. Join the circles with a line: line from $-3$ to $4$.

How do you write down the inequality shown on a number line?

e.g. open circle at $-1$ with an arrow pointing right

Read the end value, use $<$/$>$ for an open circle and $\le$/$\ge$ for a closed one, and match the sign to the arrow direction.

e.g. $x > -1$.

True or false? Circling the number $4$ on a number line is enough to show $x < 4$.

False. You also need the circle unshaded and a line with an arrow going left.

True or false? When you write down the inequality from a number line, an arrow pointing right means the sign is $>$ or $\ge$.

True. The solutions are the values greater than the circled number.

How do you solve a one-step linear inequality?

e.g. $x + 7 > 10$

Do the same inverse operation to both sides, keeping the inequality sign.

e.g. subtract 7: $x > 3$.

What are the steps to solve a two-step linear inequality?

e.g. $7x - 27 < 8$

1. Add or subtract to isolate the term in $x$: $7x < 35$.
2. Divide both sides by the coefficient: $x < 5$.
3. Write the answer as an inequality, not an equation: $x < 5$.

What are the steps to solve a linear inequality with the unknown on both sides?

e.g. $3x + 5 \ge x + 17$

1. Collect the $x$ terms on one side: $2x + 5 \ge 17$.
2. Collect the numbers on the other side: $2x \ge 12$.
3. Divide by the coefficient of $x$: $x \ge 6$.

What is the first step in solving a linear inequality containing brackets?

e.g. $2(x + 3) \le 10$

Expand the brackets.

e.g. $2x + 6 \le 10$, giving $x \le 2$.

True or false? An answer of $x = 6$ is an acceptable solution to the inequality $3x + 5 \ge x + 17$.

False. The solution set must be written as an inequality: $x \ge 6$.

After solving an inequality, how do you find the greatest integer that satisfies it?

e.g. $5y - 7 < 16$ gives $y < 4.6$

Take the largest whole number below the critical value (or equal to it if the sign is $\le$).

e.g. $4$.