Graphing Linear Inequalities

Graphing Linear Inequalities exercise

• Decide which graph describes the given situation.

  Hints

  First, have a look at the line. Does the line correspond to the inequality $y=-1.5x+5$?

  For $\ge$ or $\le$ situations, the line is solid otherwise it needs to be dashed.

  • $\ge$ or $>$ will give us solutions above and
  • $\le$ or $<$ beyond the line.

  Solution

  The inequality

  $y\ge -1.5x+5$

  is written in slope-intercept form.

  To graph, draw the line of the equation $y=-1.5x+5$ on a coordinate grid :

  • this line passes the y axis at $y=5$ and
  • the slope is $m=-1.5$. So for each one step to the right we go 1.5 steps down.

  Because we have a greater than or equal to situation

  • the line is solid, and
  • the solutions lie above the solid line.

  So the graph on the right matches the inequality.

• Determine which graph corresponds to which inequality.

  Hints

  First, take a look at the lines.

  Are the lines solid ($\le$ or $\ge$) or dashed ($<$ or $>$)?

  Solution

  Graphing linear inequalities:

  1. First, we draw the line corresponding to the equation $y=mx+b$.
  2. Next, we look at the inequality sign:

  • $\ge$ $\rightarrow$ solid line and the solutions lie above the line
  • $>$ $\rightarrow$ dashed line and the solutions lie above the line
  • $\le$ $\rightarrow$ solid line and the solutions lie beyond the line
  • $<$ $\rightarrow$ dashed line and the solutions lie beyond the line

  There are graphs that couldn't be matched up:

  • the violet one belongs to the equation $y=-1.5x+6$ which is not given and
  • the turquoise one belongs to the inequality $y<x+2$.

  These were the correct matches:

  1. The red graph belongs to the inequality $y\ge -1.5x+5$.
  2. The green graph belongs to the inequality $y\le x+2$.
  3. The blue graph belongs to the inequality $y> -\frac12x+5$.
  4. The orange graph belongs to the inequality $y<-x+4 $.

• Decide which values would earn aunt Sally enough money.

  Hints

  $x$ is the price for one cookie, and $y$ represents the price for one glass of lemonade.

  Plug the prices into the inequality:

  $15\times x+10\times y\ge 50$

  and check if Aunt Sally met her goal.

  Remember that the inequality shown is greater than or equal to.

  Solution

  Each combination of the prices for the cookies and lemonade can be drawn as a point on the coordinate graph. Just have a look at the picture on the right.

  The price for one cookie is represented by $x$, and the price for one glass of lemonade is represented by $y$. Therefore, you have to look at the x and y axis to find a suitable combination.

  $\mathbf{x=2,~y=2}$. You can see this point is plotted on the line:

  $15\times 2+10\times 2=30+20=50$. Aunt Sally needs to earn $50, so she is satisfied with her earnings.

  $\mathbf{x=1.2,~y=2.5}$. This point below the line. Therefore, it doesn't meet aunt Sally's goal of earning at least $\$50$. $15\times 1.2+10\times 2.5=18+25=43<50$.

  $\mathbf{x=3,~y=1.5}$. You can see this point lies above the line:

  $15\times 3+10\times 1.5=45+15=60$. This is more than the $\$50$ Aunt Sally had hoped to earn. She will be very satisfied with her earnings.

• Decide which graph corresponds to the given inequality.

  Hints

  First, write the inequality.

  You'll get: $12\times x+10\times y\ge 30$.

  Rewrite this inequality into an equation in slope-intercept form.

  The equation describing the line is:

  $y=-1.2x+3$.

  Solution

  Let's have a look at the inequality :

  $12\times x+10\times y\ge 30$.

  We rewrite this inequality as an equation in slope-intercept form: $\begin{array}{rcl} 12 x+10 y&\ge& 30\\ \color{#669900}{-12x} && \color{#669900}{-12x}\\ 10y&\ge&-12x+30. \end{array}$

  Now we divide by $10$: $y\ge -1.2x+3$.

  We draw theline corresponding to $y=-1.2x+3$.

  • Because of the $\ge$ sign this line is solid and
  • because of greater or equal to sign the solutions lie above this line.

  The correct graph can be seen on the right.

• Summarize how you can recognize the solution depending on the inequality sign.

  Hints

  If the points on the line are also solutions, you can recognize this by a solid line otherwise a dashed line.

  If the points on the line are also solutions, the corresponding inequality sign is $\ge$ or $\le$.

  Greater corresponds to above the line.

  Solution

  Just remember the solutions depending on the relation sign:

  • $\ge$ $\rightarrow$ solid line and the solutions lie above the line.
  • $\le$ $\rightarrow$ solid line and the solutions lie beyond the line.
  • $>$ $\rightarrow$ dashed line and the solutions lie above the line.
  • $<$ $\rightarrow$ dashed line and the solutions lie beyond the line.

  Or:

  • Greater than or less than as well as equal $\rightarrow$ solid line and dashed line otherwise.
  • Greater than or greater than or equal to $\rightarrow$ above the line and beyond otherwise.

• Decide if the given points lie within the area described by the inequalities.

  Hints

  Draw the lines corresponding to $y=x+3$ and $y=x+1$ into a coordinate system.

  Hint: these lines are parallel.

  There's no point that's a solution to both inequalities.

  Pay attention to the different signs of the inequalities.

  Solution

  The shaded areas of the graph on the right shows the solution for two inequalties.

  The solid red line belongs to the equation $y=x+1$, the dashed one corresponds to $y=x+3$. These lines are parallel.

  The solutions of $y\le x+1$ lie in the red area, and those of $y>x+3$ in the green area. Remember that the points on the green line don't belong to the solution set.

  So the point on the green line as well as both points between the two lines don't belong to either inequality.

  The point $(0.5,4.5)$ are a solution to the inequality $y>x+3$.

  All the other points, including $(1,2)$ lying on the red line, are solutions to the inequality $y\le x+1$.