Solving Systems of Inequalities by Graphing

Solving Systems of Inequalities by Graphing exercise

• Explain how to solve a system of linear inequalities by graphing.

  Hints

  Keep in mind

  • $\le$ or $\ge$ $\rightarrow$ solid line
  • $<$ or $>$ $\rightarrow$ dashed line

  Both inequalities are shown in slope-intercept form: $y=mx+b$.

  • $m$ is the slope
  • $b$ is the y-intercept

  To check if the solutions lie above or below the line formed by the graph of the inequality, pick a point and substitute the coordinates into the corresponding inequality.

  Solution

  We have the following two inequalities:

  1. $y\le -2x+1$
  2. $y<\frac32x-2$

  Start by taking a look at the inequality $\mathbf{y\le -2x+1}$:

  1. Draw the y-intercept, $1$.
  2. Next, draw the slope: move two units down and one unit to the right to find a second point on the line.
  3. Connect the dots with a solid orange line, because the inequality includes "equal to".
  4. Shade in the area of the solution set. The solutions lie below the orange line because the inequality includes "less than".

  We proceed in the same way with the second inequality, $\mathbf{y<\frac32x-2}$:

  1. Draw the y-intercept, $-2$.
  2. Next, draw the slope: move up three units and two units to the right.
  3. Connect the dots with a green dashed line, because of the less than sign.
  4. Shade in the area of the solution set. The solutions lie under the green line.

  The set of all possible solutions of the two inequalities combined is given by the intersection of the shaded areas.

• Find all solutions to Adventure Mike's riddle.

  Hints

  $\le$ or $\ge$ inequalities are represented with solid lines while $<$ or $>$ inequalities are represented with dashed lines.

  Dont forget:

  • $\le$ or $<$ $\rightarrow$ the solutions lie below the line
  • $\ge$ or $>$ $\rightarrow$ the solutions lie above the line

  For equations written in the slope-intercept form $y=mx+b$, the y-intercept is represented by the term $b$.

  Solution

  Here you can see the correct graph.

  $\mathbf{y\le -2x+1}$

  • the orange solid line passes through the y-intercept, $1$
  • the solutions lie below the line

  $\mathbf{y<\frac32 x-2}$

  • the green dashed line passes through the y-intercept, $-2$
  • the solutions lie below the line

  Keep in mind:

  • draw solid lines for $\le$ or $\ge$ inequalities
  • draw dashed lines for $<$ or $>$ inequalities

  Check any point to verify the shading of the solution set:

  • $\le$ or $<$ $\rightarrow$ below the line
  • $\ge$ of $>$ $\rightarrow$ above the line

• Determine where the party will take place.

  Hints

  You can pick some points and plug them into the inequality to check if they belong to the solution set.

  The lines formed by $\le$ or $\ge$ inequalities are part of the solution set.

  The lines formed by $<$ or $>$ inequalities aren't part of the solution set.

  Solution

  Let's start with the inequality $\mathbf{y\ge 2x-2}$:

  1. The y-intercept is $-2$.
  2. Move two units up and one unit to the right to plot additional points.
  3. Connect the points with a solid line, since they are formed by a greater than or equal to inequality. The points on the line are included in our solution set. This inequality is shown by the green line.
  4. The green shaded area is the solution set for this inequality.

  Now we graph $\mathbf{y<-\frac12x+3}$:

  1. The y-intercept is $3$.
  2. Move one unit down and two units to the right to plot additional points.
  3. Connect the points with a dashed line, since they are formed by a less than inequality. This is shown by the orange line.
  4. The orange shaded area is the solution set for this inequality.

  The set of all solutions satisfying both inequalities is given by the intersection of the shaded areas. Don't forget that points on the green line of $\mathbf{y\ge 2x-2}$ also belong to the solution set.

• Decide if the location of the treasure lies inside the area where the robbers are searching.

  Hints

  Each point in the coordinate system is given by $P(p_x,p_y)$, where $p_x$ is the x-coordinate, and$p_y$ is the y-coordinate of the point $P$.

  Plug each of the given points into all three of the inequalities. The solution must satisfy each of the inequalities.

  Pay attention to the $<$ sign in the second inequality.

  Are points that are exactly on the line part of the solution set?

  Draw the graphs of the inequalities and see if the coordinates are within the intersection area.

  Solution

  You can solve a system of three inequalities by graphing in a similar way used for two inequalities:

  1. Determine the y-intercept
  2. Use the slope to plot additional points
  3. Connect the points with a solid or dashed line depending on the inequality sign
  4. Shade the area belonging to the solution set, either below or above the line, depending on the inequality sign
  5. The intersection area of all shaded areas is where the hidden treasure can be found

  In the graph on the right, you can see the system of the three equations.

  We can use this graph for our decision:

  • Crown $(0,3)$ lies on the green and on the red line. But pay attention: the red line is dashed, which means the points on the line aren't included in the solution set. So the robbers can't find this specific treasure.
  • Coins $(0,2)$ lie inside the solution area. You can see it directly on the graph.
  • Necklace $(2,1.75)$ lies inside the solution area.
  • Antique sword $(-1,1)$ lies inside the solution area.
  • Bottle of love potion $(2,3)$ lies outside the solution area. When we plug in our y-coordinate, $3$, we see that the inequality is not true: $3\not < \frac14\times 2+3=3.5$

• Describe how to determine the solution set for the inequality.

  Hints

  An equation in slope-intercept form is: $y=mx+b$

  • $m$ is the slope
  • $b$ is the y-intercept

  For all $\le$ or $\ge$ inequalities, draw a solid line and for all others, draw a dashed line.

  Take any point $P(x_p,y_p)$ and check if $x_p$ and $y_p$ are in the solution set of the given inequality. This helps you verify if the solutions lie above or below the line formed by the inequality.

  Solution

  Here you can see the graph of $y<\frac32x-2$:

  • the y-intercept is $-2$
  • for the slope $m=\frac32$, move up three units and right two units
  • connect the points $(0,-2)$ and $(2,1)$ with a dashed line
  • all solution sets lie below the dashed line

• Assign the inequality.

  Hints

  The y-intercept is easily identified in the coordinate system. This is the place where the line crosses the y-axis.

  You can calculate the slope by moving up or down and right. For example $m=-\frac34$:

  • move three units down (because of the negative sign)
  • and move four units to the right

  Keep in mind:

  • $\ge$ or $\le$ $\rightarrow$ solid line
  • $>$ or $<$ $\rightarrow$ dashed line

  How do you know if the solution area lies above or below the line formed by the inequality?

  • $\ge$ or $>$ $\rightarrow$ above
  • $\le$ or $<$ $\rightarrow$ below

  Solution

  Let's reconstruct the three inequalities:

  Start with the red line and the corresponding shaded area:

  1. The y-interecept is $-3$.
  2. We move three units up and five units to the right. This gives us the slope $m=\frac35$.
  3. The line is solid and the shaded area is above the line.

  We can write the inequality:

  $y\ge\frac35x-3$

  $~$

  Now let's have a look at the green line:

  1. The y-interecept is $3$.
  2. We move up three units and one unit to the right. This gives us the slope $m=3$.
  3. The line is dashed and the shaded area is below the line.

  We can write the inequality:

  $y<3x+3$

  $~$

  Finally, we examine the blue line:

  1. The y-interecept is $4$.
  2. We move two units down and one unit to the right. This gives us the slope $m=-2$.
  3. The line is solid and the shaded area is above the line.

  We can write the inequality:

  $y<-2x+4$