Compound Inequalities

Compound Inequalities exercise

• Write an inequality to describe the situation.

  Hints

  The phrases less than, at most, between, more than, or at least have special meanings in math.

  For example, more than means $>$. The equal sign is not included.

  At most $10$ means $\le 10$ and $10$ is included.

  Solution

  Jerry and Larry want to ride the rollercoaster. They have a problem with the minimum and maximum height allowed to ride the roller coaster.

  • At least $4$ feet $7$ inches can be expressed mathematically using $\ge$. So $x\ge 4$ feet $7$ inches or $4$ feet $7$ inches $\le x$.
  • Shorter than $6$ feet $3$ inches can be expressed as $x<6$ feet $3$ inches.

  Combining the two expressions, we get the compound inequality $4$ feet $7$ inches $\le x<6$ feet $3$ inches.

• Simplify the following inequalities.

  Hints

  To solve an inequality treat it like an equation and isolate the variable $x$ using opposite operations.

  Remember, when multiplying with or dividing by a negative value, flip the sign!

  Use Opposite Operations:

  • Multiplication ($\times~\longleftrightarrow~\div$)
  • Division ($\div~\longleftrightarrow~\times$)
  • Addition ($+~\longleftrightarrow~-$)
  • Subtraction ($-~\longleftrightarrow~+$)

  Solution

  Let's have a look at solving compound inequalities.

  We start with a compound inequality: $4<2x<10$.

  We can treat this inequality just like an equation by using opposite operations to isolate the variable $x$.

  REMEMBER! If we multiply or divide both sides of the equation by a negative value, we have to flip the sign.

  The left side divided by $2$ gives us $2<x$, and the right side becomes $x<5$. Because these inequalities are connected by the word and, we get the following compound inequality: $2<x<5$.

  Next we look at $3x-1<14$ and $4x-2>46$.

  First we transform the first inequality:

  • Adding $1$ on both sides gives us $3x<15$
  • Dividing by $3$ leaves us with $x<5$. Notice that the sign is still pointing in the same direction.

  Now for the second inequality:

  • Add $2$ to both sides to get $4x>48$
  • Dividing by $4$ on both sides gives us our solution, $x>12$.

  Together, we get the compound inequality $x<5$ or $x>48$.

  Let's have a look at the difference between and and or.

  And means that both inequalities must be fulfilled. This can be seen as the intersection of two sets.

  Or is a disjunction. This means that at least one of the inequalities has to be fulfilled.

• Decide which number line belongs to which compound inequality.

  Hints

  First, isolate the $x$.

  • $<x$ less than, $x$ is not included
  • $\le x$ at most, $x$ is included
  • $>x$ greater than, $x$ is not included
  • $\ge x$ at least, $x$ is included

  Solution

  We can represent compound inequalities with number lines. We can indicate included numbers with filled circles and excluded numbers with empty circles.

  First, we have to isolate the variable $x$ in each inequality.

  $\begin{array}{rclcrccl} 3&<&3x&<&12&&| (\times~\leftrightarrow~\div) \text{ Opposite Operations}\\ 1&<&x&<&4 \end{array}$

  This is the blue number line with two empty circles $4$ and $14$.

  $\begin{array}{rclcrccl} 4&<&3x+1&\le&13&&| (+~\leftrightarrow~-) \text{ Opposite Operations}\\ 3&<&3x&\le&12&&| (\times~\leftrightarrow~\div) \text{ Opposite Operations}\\ 1&<&~~x&\le&4 \end{array}$

  This is the green line number with an empty circle at $1$ and $4$.

  $\begin{array}{rclcrccl} 3&\le&2x-5&\le& 9&&| (-~\leftrightarrow~+) \text{ Opposite Operations}\\ 8&\le&2x&\le&14&&| (\times~\leftrightarrow~\div) \text{ Opposite Operations}\\ 4&\le&~~x&\le &7 \end{array}$

  This is the yellow line number.

  $\begin{array}{rclcrccl} 5&<&3x-4&\le& 11&&| (-~\leftrightarrow~+) \text{ Opposite Operations}\\ 9&<&3x&\le&15&&| (\times~\leftrightarrow~\div) \text{ Opposite Operations}\\ 3&<&~~x&\le&5 \end{array}$

  This is the red line number.

• Solve the inequality.

  Hints

  To solve an inequality treat it like an equation, but don't forget to consider the sign.

  Remember, if you multiply or divide by a negative value, you have to flip the sign!

  Or more explicitly:

  • $<$ changes to $<$, and vice versa
  • $\le$ changes to $\ge$, and vice versa

  Solution

  We can solve inequalities by following these simple steps:

  • Use opposite operations
  • If you multiply or divide by a negative number, flip the sign
  • Isolate the variable $x$

  $\begin{array}{rclcrcl} -10&\le& -2\times(x+1)&<&-4&~&|\text{ Distributive Property}\\ -10&\le& -2x-2&<&-4\\ \color{#669900}{+2}&&\color{#669900}{+2}&&\color{#669900}{+2}&~&| (-~\leftrightarrow~+) \text{ Opposite Operations}\\ -8&\le& -2x&<&-2&~&|\div(-2)\\ \color{#669900}{\div(-2)}&&\color{#669900}{\div(-2)}&&\color{#669900}{\div(-2)}&~&| (\times~\leftrightarrow~\div) \text{ Opposite Operations}\\ 4&\ge& x&>&1&~&|\text{ Flip the Sign(s)} \end{array}$

  This compound inequality can also be written as follows:

  $1<x\le4$.

• Explain what the inequality means.

  Hints

  For example, more than means $>$.

  $\ge x$ means at least, $x$ is included

  For example, less than means $<$.

  $\le x$ means at most, $x$ is included

  $2\le x\le4$ means that $x$ is $2$ or more as well as $4$ or less.

  Solution

  First let's have a look at the signs and their meanings:

  • $<x$ less than, $x$ is not included
  • $\le x$ at most, $x$ is included
  • $>x$ greater than, $x$ is not included
  • $\ge x$ at least, $x$ is included

  The above compound inequality tells us that visitors who want to ride the roller coaster have to be at least $4$ feet and $7$ inches tall and shorter than $6$ feet $3$ inches. If a visitor is $6$ feet $3$ inches tall, he or she cannot ride the roller coaster.

• Determine the opposite inequality.

  Hints

  The opposite of inclusion (and) is exclusion (or).

  The opposite of

  • $<$ is $\ge$
  • $\le$ is $>$

  ...and vice versa.

  Solution

  The opposite of inclusion (and) is exclusion (or).

  If we want to examine all values for $x$ that don't satisfy the compound inequality $12\le3\times(x+2)<24$, we first have to isolate the variable $x$:

  $\begin{array}{rcrcl} ~~\\ 12&\le& 3\times(x+2)&<&24&&|\text{ Distributive Property}\\ 12&\le& 3x+6&<&24\\ \color{#669900}{-6}&&\color{#669900}{-6}&&\color{#669900}{-6}&&| (+~\leftrightarrow~-) \text{ Opposite Operations}\\ 6&\le& 3x&<&18\\ \color{#669900}{\div3}&&\color{#669900}{\div3}&&\color{#669900}{\div3}&&| (\times~\leftrightarrow~\div) \text{ Opposite Operations}\\ 2&\le& x&<&6\\ ~~\\ \end{array}$

  But what does this inequality mean? Right, both inequalities, $2\le x$ and $x<6$ have to be satisfied.

  The opposite of the first inequality is $2>x$ or $x<2$. The opposite of the second inequality is given by $6\le x$.

  The opposite of and is or. So the resulting compound inequality is

  $x<2$ or $x\ge 6$.