Solving Systems of Inequalities

Solving Systems of Inequalities exercise

• Establish the corresponding system of inequalities.

  Hints

  Make sure Caleb can make use of as many nails and wooden boards as possible,

  but not more.

  To check if Caleb could build enough shelves for 5 baseball caps and 19 pairs of sneakers, plug those values into both of the inequalities.

  Solution

  Caleb is very proud of his collection of sneakers and baseball caps. To show off his collection, he would like to build a shelving unit for them.

  First let's assign variables to the unknown number of baseball caps and sneakers Caleb will be able to store: Let $b$ represent the number of baseball caps and $s$ the number of pairs of sneakers.

  Caleb has $150$ nails. He needs $5$ nails per baseball cap and $5$ nails per pair of sneakers. Thus we get $5b+5s\le150$.

  The number of wooden boards Caleb has is $66$. He needs $2$ boards per baseball cap and $3$ per pair of sneakers. So we get $2b+3s\le66$.

• Identify the area containing all solutions to the system of inequalities.

  Hints

  First we replace $b$ by $x$ and $s$ by $y$ to get the following system of inequalities:

  $\begin{array}{rcl} 5x+5y&\le&150\\ 2x+3y&\le&66 \end{array}$

  Next we write each inequality in slope intercept form:

  Draw the boundary line of each inequality and check which combinations of $x$ and $y$ satisfy it.

  The intersection of both solution sets, including the boundary lines, is the solution set to the system of inequalities.

  Solution

  We can solve the system of inequalities by graphing in the following manner:

  • First we replace the variables $b$ and $s$ with $x$ and $y$.

  $\begin{array}{rcl} 5x+5y&\le&150\\ 2x+3y&\le&66 \end{array}$

  $~$

  • Next we write each inequality in a slope intercept form.

  $\begin{array}{rclll} 5x+5y&\le&150\\ \color{#669900}{-5x}&&\color{#669900}{-5x}\\ 5y&\le&-5x+150\\ \color{#669900}{\div 5}&&\color{#669900}{\div 5}\\ y&\le&-x+30 \end{array}$

  and

  $\begin{array}{rclll} 2x+3y&\le&66\\ \color{#669900}{-2x}&&\color{#669900}{-2x}\\ 3y&\le&-2x+66\\ \color{#669900}{\div 3}&&\color{#669900}{\div 3}\\ y&\le&-\frac23x+32 \end{array}$

  $~$

  • Finally, draw the boundary lines $y=-x+30$ (in red) and $y=-\frac23x+32$ (in blue). Because we have a $\le$ relation for each equation, the area of the solution set for each inequality lies under each line. The solution set of the system of inequalities is the intersection of both solution sets, including the boundary lines.

• Determine if the the number of shelves needed could be constructed.

  Hints

  First establish the inequality for the number of nails as well as the inequality for the number of wooden boards.

  If the resulting number of nails for the given amount of baseball caps and pairs of sneakers is less than or equal to $400$, then the inequality for nails is fulfilled.

  Solution

  For Caleb's new shelves, he needs $10$ nails and $4$ wooden boards per baseball cap and $12$ nails and $5$ boards per pair of sneakers. He has $400$ nails and $160$ wooden boards at disposal.

  Let $x$ represent the number of baseball caps and $y$ represent the number of pairs of sneakers. We get the following system of inequalities:

  $(1)~~10x+12y\le 400$ for the nails,

  and

  $(2)~~4x+5y\le 160$ for the wooden boards.

  $~$

  To check if Caleb could construct shelves for $10$ baseball caps and $20$ pairs of sneakers, we put these values into our system of inequalities:

  $(1)~~10(10)+12(20)=340\le400$,

  and

  $(2)~~4(10)+5(20)=140\le160$.

  We can now see that Caleb can construct enough shelves for $10$ baseball caps and $20$ pairs of sneakers.

• Identify which graph belongs to which system of inequalities.

  Hints

  The blue and red line are the boundary lines: if we have a $\le$ relation, then all solutions lie under the boundary line.

  If you have two inequalities, then both must be satisfied.

  The solution set of two inequalities is the intersection of both solution sets.

  Solution

  How can we graph the solution set of a system of inequalities?

  1. First we draw the boundary lines of the solutions sets.
  2. For a $\le$ relation, all solutions lie under the boundary line. For a $\ge$ relation, all solutions lie over the boundary line.
  3. Draw lines for $\le$ or $\ge$ relations and dashed lines stand for $<$ or $>$ relations.
  4. The intersection of the solution sets of two inequalities is the solution set of the system of inequalities.

  $~$

  Now let's have a look at the graphs from the left to the right:

  The first graph is given by

  $\begin{array}{rcl} 5x+3y&\ge&15\\ x+3y&\le&6 \end{array}$

  The second graphing is given by

  $\begin{array}{rcl} 5x+3y&\le&15\\ x+3y&\le&6 \end{array}$

  The third graph is given by

  $\begin{array}{rcl} 5x+3y&\le&15\\ x+3y&\ge&6 \end{array}$

  And the last, but not least, graph is given by

  $\begin{array}{rcl} 5x+3y&\ge&15\\ x+3y&\ge&6 \end{array}$

• Write each inequality in slope intercept form.

  Hints

  Slope intercept form is given by $y\le mx+b$, where $m$ is the slope and $b$ the $y$-intercept.

  You can subtract from both sides:

  Keep in mind: You have to change the relation sign if you multiply or divide by a negative number.

  Solution

  To get write inequality in slope intercept form, $y\le mx+b$, we perform the following operations:

  $\begin{array}{rclll} 5x+5y&\le&150\\ \color{#669900}{-5x}&&\color{#669900}{-5x}\\ 5y&\le&-5x+150\\ \color{#669900}{\div 5}&&\color{#669900}{\div 5}\\ y&\le&-x+30 \end{array}$

  and

  $\begin{array}{rclll} 2x+3y&\le&66\\ \color{#669900}{-2x}&&\color{#669900}{-2x}\\ 3y&\le&-2x+66\\ \color{#669900}{\div 3}&&\color{#669900}{\div 3}\\ y&\le&-\frac23x+22 \end{array}$

• Decide what combinations of comic and fantasy books Caleb could buy.

  Hints

  First establish the system of inequalities.

  Pay attention to signs.

  You can graph the solution set.

  Solution

  First we assign $x$ to the number of comics and $y$ to the number of fantasy books.

  The total number of books is the sum of each number. So we get

  $x+y\ge 5$.

  Because Caleb has $24$ dollars, we get the second inequality,

  $3x+5y\le 24$.

  For getting the solution set we can draw the boundary lines. The intersection of the solution sets is the solution set of the system of inequalities. Last we can check each point in the coordinate system.

  The only combinations which don't lie in the solution set are

  • 2 comics and 4 fantasy books,
  • 5 comics and 2 fantasy books, and
  • 3 comics and 1 fantasy book.