Solving Systems of Inequalities by Graphing 03:51 minutes

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Transcript Solving Systems of Inequalities by Graphing

Frank, the insurance man, is on a flight to Peru to track down Adventure Mike in the Peruvian jungle. Adventure Mike needs to renew his Ultra Danger Insurance Package. Otherwise, he will not have any insurance on his various adventures.

The system of linear inequalities

Frank takes out his phone to look at the text message that he received a few days ago from Adventure Mike. The text message provides a hint as to which region in the jungle Adventure Mike is searching for ancient relics. Frank knows that Adventure Mike likes to communicate in riddles. He recognizes Mike's missive as a system of linear inequalities.

Good thing Frank knows how to solve a system of linear inequalities by graphing. Let's see how Frank solves the system of linear inequalities. Here's a map of the surrounding region. Since we have a system of linear inequalities, we need to include the x- and y-axes in order to graph them. Our starting point, or the origin, is the city of Cuzco.

We need to graph the system of inequalities. Each inequality is written in slope-intercept form. For the first inequality, 1 is the y-intercept or the ordered pair (0, 1). To find the second point, we use the slope of the line. The slope of the line is -2 or -2 over 1. We move down 2 from the y-intercept and right 1. The resulting ordered pair is (1, -1).

Less than or Equal to

Since the inequality represents 'Less than or Equal to', the graph of the inequality is a solid line. All of the points to the left of the inequality are true. We can shade to the left of the inequality line. For the second inequality, 2 is the y-intercept, or the ordered pair (0, -2). To find the second point, we use the slope of the line.

Less than

The slope of the line is 3 over 2. We move up 3 from the y-intercept and right 2. The resulting ordered pair is (2, 1). Since the inequality represents 'Less than', the graph of the inequality is a dotted line. Since all of the points to the right of the inequality are true. We can shade to the right of the inequality line.

The solution to the system of inequalities is where the shading from each inequality overlaps. Frank knows in which part of the jungle to look, so he sets off to find Adventure Mike. Wow! Frank finds an ancient Incan temple. But, what is carved into the side of it? Adventure Mike has left another hint!

Frank pulls out his map again so he can crack this clue and narrow down the possible whereabouts of Adventure Mike. We need to graph the inequality. The inequality is written in slope-intercept form. For the first inequality, -4 is the y-intercept or the ordered pair (0, -4). To find the second point, we use the slope of the line. The slope of the line is 1 over 4. We move up 1 from the y-intercept and right 4. The resulting ordered pair is (4, -3).

Greater than or Equal to

Since the inequality represents 'Greater than or Equal to', the graph of the inequality is a solid line. All of the points above the inequality are true. We can shade above the inequality line. The solution to the system of inequalities is where the shading from each inequality overlaps. Now Frank knows exactly where to go...Oh, so THAT'S why Adventure Mike sent those strange messages in the first place.

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

    Tipps

    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.

    Lösung

    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.

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

    Tipps

    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.

    Lösung

    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

  • Find all solutions to Adventure Mike's riddle.

    Tipps

    $\le$ 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 $b$.

    Lösung

    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

  • Assign the inequality.

    Tipps

    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

    Lösung

    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:

    $y<-2x+4$

  • Determine where the party will take place.

    Tipps

    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.

    Lösung

    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.

    Tipps

    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.

    Lösung

    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: $3\not < \frac14\times 2+3=3.5$