2.15.2 Interpreting graphical inequalities
To interpret inequalities on a graph
- Identify the boundary line of the inequality on the graph.
- This line is usually solid or dashed, depending on whether or not the inequality includes an equal sign.
- Determine which side of the boundary line represents the solution to the inequality.
- Check if the solution to the inequality includes the boundary line.
- If the boundary line is solid, the solution includes the line itself.
- If the boundary line is dashed, the solution does not include the line.
- Identify the direction of the inequality.
- If the shaded region is
- above the line (for an inequality in the form \(y > mx + b\))
- to the right of the line (for an inequality in the form \(x > c\))
- below the line (for an inequality in the form \(y < mx + b\))
- to the left of the line (for an inequality in the form \(x < c\))
- If the shaded region is
- If the inequality is a system of two or more inequalities, the solution is the region where all of the shaded regions overlap.
- This region is the set of points that satisfy all of the inequalities in the system.
- Finally, label the shaded region with the appropriate inequality or system of inequalities that it represents.
- This will help you keep track of which region corresponds to which inequality or system of inequalities.
Worked example:
Write down the three inequalities that defines the region R .
Identify the boundary line of the inequality on the graph.
Label the lines as l, m and n
The line l is a straight horizontal line where all the y-coordinates are equal to 1.5
\(y=1.5\)
The gradient of line m is 2 (2 units up and 1 unit to the right) and y-intercept is (0,1)
\(y=2x+1\)
The gradient of line n is -1 (1 unit down for every unit to the right) and y-intercept is (0, 4)
\(y=-x+4\)
\(y=4-x\)
Identify the direction of the inequality
\(y≥1.5\)
\(y≤2x+1\)
\(y<4-x\)
Test yourself
Question 1:
By shading the unwanted regions of the grid, find and label the region R that satisfies the following inequalities
\(y≤5,\ 2x+y≥6,\ yx+1\)
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Question 2:
Raheem makes baskets and mats.
Each week he makes x baskets and y mats.
He makes fewer than 10 mats.
The number of mats he makes is greater than or equal to the number of baskets he makes.
One of the inequalities that shows this information is \(y<10\).
Write down the other inequality.
[1]
Raheem takes 214 hours to make a basket and 112 hours to make a mat. Each weak he works for a maximum of 22.5 hours.
Show that \(3x+2y≤30\).
[2]
On the grid, draw three straight lines and shade the unwanted regions to show these inequalities.
[5]
- Raheem makes $40 profit on each basket he sells and $28 profit on each mat he sells.
Calculate the maximum profit he can make each week.
[2]
Question 3:
A car hire company has x small cars and y large cars.
The company has at least 6 cars in total.
The number of large cars is less than or equal to the number of small cars.
The largest number of small cars is 8.
- Write down three inequalities, in terms of x or y, to show this information.
[3]
- A small car can carry 4 people and a large car can carry 6 people.
One day, the largest number of people to be carried is 60.
Show that \(2x+3y≤30\).
[1]
- By shading the unwanted regions on the grid, show and label the region R that satisfies
all four inequalities.
[6]
- Find the number of small cars and the number of large cars needed to carry exactly 60 people
[1]
- When the company uses 7 cars, find the largest number of people that can be carried.
[2]
Question 4:
Find the three inequalities that define the region R .
[4]
Find the point x, y, with integer coordinates, inside the region R such that \(3x+5y=35\) .
[2]
Question 5:
Klaus buys x silver balloons and y gold balloons for a party.
He buys
- more gold balloons than silver balloons
- at least 15 silver balloons
- less than 50 gold balloons
- a total of no more than 70 balloons
- Write down four inequalities, in terms of \(x or y\), to show this information.
[4]
- On the grid, show the information from part a by drawing four straight lines and
shading the unwanted regions.
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When a quantity exponentially decays, it decreases by \( r \)% per year for \( t \) years from an initial amount, \( P \).
- Real-world examples of exponential decay include hot water cooling, the value of a car decreasing over time, and radioactive decay
- The same formula from depreciation is used
\( A=P(1- \frac{r}{100})^t \)
Where
- \( A \)= final amount of the quantity
- \( P \)= principal amount of the quantity
- \( r \)= percentage decrease
- \( t \)= time period
Worked example:
The population of a village decreases exponentially at a rate of \( 4 \)% each year.
The population is now \( 255 \).
Calculate the population \( 16\) years ago.
Substitute \( A=255 \) , \( r=4 \) and \( t=16 \) in the formula \( A=P(1- \frac{r}{100})^t \) to find \( P \).
\( 255=P(1- \frac{4}{100})^{16} \)
Rearrange to make \( P \) the subject
\( P= \frac{255}{{(1- \frac{4}{100})}^{16}} \)
\( P=490.0049325 \)
The answer is a number of people, so we need to round the answer to the nearest whole number.
\( 490 \)
The population was \( 490\).
Test yourself
Question 1:
The population of a city is increasing exponentially at a rate of \( 2 \)% each year.
The population now is \( 256000 \).
Calculate the population after \( 30 \) years.Give your answer correct to the nearest thousand.
[3]
Question 2:
Over a period of \( 3 \) years, a company’s sales of biscuits increased from \( 15.6 \) million packets
to \( 20.8 \) million packets.
The sales increased exponentially by the same percentage each year.
Calculate the percentage increase each year.
[3]
Question 3:
Mohsin’s earnings increase exponentially at a rate of \( 8.7 \)% each year.
During \( 2018 \) he earned \( $ \ 195600 \).
During \( 2027 \), how much more does he earn than during \( 2018 \)?
[3]
Question 4:
In a city the population is increasing exponentially at a rate of \( 1.6 \)% per year.
Find the overall percentage increase at the end of \( 20 \) years.
[2]
Question 5:
Over a period of \( 3 \) years, a company’s sales of biscuits increased from \( 15.6 \) million packets to
\( 20.8 \) million packets.
The sales increased exponentially by the same percentage each year.
Calculate the percentage increase each year.
[3]