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| 10 Theory slides |
| 9 Exercises - Grade E - A |
| Each lesson is meant to take 1-2 classroom sessions |
Here are a few recommended readings before getting started with this lesson.
When graphing a linear equation, the resulting graph is a line.
The graph looks this way because the solution set of a linear equation forms a straight line. How would the graph change if the equals sign is replaced with an inequality symbol?A linear inequality is an inequality that represents a linear relationship involving one or more variables. Therefore, the exponent of all variables is 1. As with any other inequality, linear inequalities can be strict or non-strict.
Strict Linear Inequalities | Non-Strict Linear Inequalities |
---|---|
-13>-2x+4y | 7x−y≥21 |
-9y<17+x | 9x+3y≤6 |
Linear inequalities are similar to linear equations. The difference is that, while the solutions to linear equations are represented by all the points that lie on a line, the solution set to a linear inequality is represented by the points on a region containing one half of the coordinate plane.
Point | Substitute | Simplify |
---|---|---|
(0,0) | 0≤3(0)+7 | 0≤7 ✓ |
(1,11) | 11≤3(1)+7 | 11≤10 × |
Verify whether the point satisfies the given linear inequality.
LHS−9x≤RHS−9x
LHS/3≤RHS/3
Write as a sum of fractions
Put minus sign in front of fraction
ca⋅b=ca⋅b
Calculate quotient
Symbol | Meaning | Type | Boundary Line |
---|---|---|---|
< | Less than | Strict | Dashed |
> | Greater than | Strict | Dashed |
≤ | Less than or equal to | Non-strict | Solid |
≥ | Greater than or equal to | Non-strict | Solid |
Therefore, in the given example, the line is solid. The boundary line can be graphed using y-intercept and slope.
If the test point is a solution to the inequality, the region that contains it must be shaded. Otherwise, the opposite region must be shaded.
In this case, the test point (0,0) is a solution to the inequality. The region containing (0,0) lies to the left of the boundary line. This is the region that must be shaded.
Stoked to go on a ski trip to the Rocky Mountains, Heichi has saved as much money as he could to buy equipment. After buying most of the equipment, he is left with $1250 to buy jackets and pants.
At his favorite ski shop, each jacket costs $150 and each pair of pants costs $200.
no morethan $1250.
No morecan be written with the less than or equal to symbol.
LHS−3x≤RHS−3x
LHS/4≤RHS/4
Write as a sum of fractions
Put minus sign in front of fraction
ca⋅b=ca⋅b
Now the given point can be tested. To do so, verify if the point lies in the shaded region.
Since the point lies outside the shaded region, the point is not a solution to the inequality.
Writing a linear inequality given a graph is similar to writing a linear equation given a graph. Two extra things must be considered.
Consider the following graph.
The procedure begins by writing the equation of the boundary line in slope-intercept form. To do so, the y-intercept and the slope must be found. Then, identify whether the inequality is strict. Finally, the inequality symbol should be determined.To write the linear equation of the boundary line, the slope and y-intercept should be identified. It is easier to do so if the shaded region is ignored.
The y-intercept is 2. The slope of the line is the quotient of the rise and run.To identify whether the inequality is strict or non-strict, the line should be considered. If it is dashed, the inequality is strict. Otherwise, the inequality is non-strict. Consider the boundary line of the given inequality.
The graph has a dashed line. Therefore, the inequality is strict. This means that the inequality sign is either < or >.
To determine the sign of the inequality, a point located in the shaded region but not on the boundary line should be tested. For simplicity, the point (1,1) will be used.
The coordinates of this point can be substituted to determine the symbol of the inequality.x=1, y=1
Identity Property of Multiplication
Add terms
Consider the following graph of a linear inequality represented on a coordinate plane.
Write the inequality in slope-intercept form.Start by writing the equation of the boundary line.
To write a linear inequality from a graph, the first step is writing the equation the boundary line. To do so, the slope and the y-intercept will be found.
The y-intercept of the line is -3. Furthermore, the slope of the line is -23, or -1.5. Using this information, the equation for the boundary line can be written in slope-intercept form.x=-3, y=-2
-a(-b)=a⋅b
Subtract term
Next, the the point (0,0) is substituted in each inequality to see if these are satisfied.
Inequality | Substitute | Simplify |
---|---|---|
y>x−3 | 0>?0−3 | 0>-3 ✓ |
y≤4x+1 | 0≤?4(0)+1 | 0≤1 ✓ |
The coordinates of the point satisfy both inequalities.
This information can be shown in the graph.
Finally, the overlapping region is the solution set for both inequalities. This means that the points in this region satisfy both inequalities at the same time.
A single inequality divides the coordinate plane in two regions, and one of those regions is the solution set of the inequality. Meanwhile, two inequalities can divide the coordinate plane in up to four regions, and sometimes there is no overlapping solutions for both inequalities at the same time.
We will evaluate the linear inequality on each ordered pair and simplify to determine which one is a solution to the inequality. If a true statement is reached, the point is in the solution set of the inequality. Let's first evaluate the inequality for ( 5, 8).
Because 8 is not less than 27, this statement is not true. Therefore, (5,8) is not a solution of the inequality. We will evaluate the inequality for the remaining points in a similar fashion.
y>5x+2 | ||
---|---|---|
Ordered Pair | Substitute | Simplify |
( 5, 8) | 9? > 5( 5)+2 | 8≠ >27 * |
( 6, 0) | 0? > 5( 6)+2 | 0≠ >32 * |
( 0, 2) | 2? > 5( 0)+2 | 2≠ >2 * |
( -1, 0) | 0? > 5( -1)+2 | 0>-3 ✓ |
Because the point (-1,0) satisfies the inequality, we can state that it is a solution to the given inequality.
Consider the following linear inequality.
What is the equation of the graph shown?There are two major steps to writing an inequality when given its graph.
Let's get started by focusing on the boundary line.
It only takes two points to create a unique equation for any line, so let's identify two points on the boundary line.
Here we have identified two points, (0, 3) and (3,4), and indicated the horizontal and vertical changes between them. This gives us the rise
and run
of the graph, which will give us the slope m.
rise/run=1/3 ⇔ m= 1/3
One of the points we selected, (0, 3), is also the y-intercept, which is great news! This means we can combine the slope m and the y-intercept at the point (0, b) to write an equation for the boundary line in slope-intercept form.
y= mx+ b ⇒ y= 1/3x+ 3
To finish forming the inequality, we need to determine the inequality symbol. This means replacing the equals sign with a blank space since it is still unknown. y ? 1/3x+3 To figure out what the symbol should be, we need a test point that lies within the solution set.
We will substitute ( 0, 0) for this test, then use the inequality symbol that suits the resulting statement.
Zero is less than 3. We can infer that, of the four inequality symbols, only two would make this a true statement, < or ≤. Returning to the given graph one last time, we can see that the boundary line is dashed, not solid. This implies that the inequality is strict. We can now form our final inequality. y< 1/3x+3 The inequality can also be written in standard form.
To graph an inequality, we should start by drawing its boundary line. We can write the equation of this line by replacing the inequality sign with an equals sign. ccc Inequality & & Boundary Line [0.8em] y> 3x+2 & & y= 3x+2 Since the line is already written in slope-intercept form, we can use its y-intercept, 2, and its slope, 3, to draw its graph. Note that we have a strict inequality, so the boundary line will be dashed.
We will use a test point to determine whether we have to shade above or below the line. If substituting this point in the inequality produces a true statement, we shade the region that contains the point. Otherwise, we shade the opposite region. For simplicity, we will use ( 0, 0) as a test point.
Since the test point produced a false statement, we will shade the region that does not contain (0,0).
We have shaded above the boundary line. Since the boundary line is dashed, the area of the solution set is above the boundary line and does not include the line. Therefore, the answer is C.
Let's start by graphing the given inequality. To do so, we must first identify the boundary line. To write the equation of the boundary line, we replace the inequality sign with an equals sign. ccc Inequality & & Boundary Line [0.4em] y ≤ 1/4x-1 & & y = 1/4x+( -1) Since the line is already written in slope-intercept form, we will use its slope 14 and its y-intercept -1 to graph it. Note that the inequality is non-strict, so the line will be solid.
Now that we have graphed the boundary line, we will determine the region to shade. To do so, we will test a point. If substituting the point into the inequality produces a true statement, we will shade the region that contains the point. Otherwise, we will shade the opposite region. For simplicity, we will test the point ( 0, 0).
Since the test point produced a false statement, we will shade the region that does not contain (0,0).
Note that the solution set contains all the points that are below the line, including the boundary line. Therefore, the region of the coordinate plane not included in the graph is the region above the line, not including the line. This can be expressed by reversing the inequality sign and making the inequality strict. y> 1/4x-1 Let's see how the graph of this inequality looks like.
Heichi is buying food for a party. A pizza costs $6 and a bottle of soda costs $2. Which graph represents the number of pizzas and sodas that Heichi can buy if he has $30?
First, we will write an expression for Heichi's purchases. To do so, we can assign variables to the number of pizzas and sodas that Heichi can buy. We can call the number of pizzas he buys p and the number of bottles of sodas s. The total price of his purchases can be written as an expression. 6p + 2s Since Heichi can spend at most $30, we know that the total purchases have to be less than or equal to 30. 6p + 2s ≤ 30 We can see that this is a linear inequality. We might have an easier time graphing the inequality by writing it in slope-intercept form first.
We can see that the slope is - 13, while the y-intercept is 5. We can use this information to graph the boundary line of the inequality.
Note that we should only consider the first quadrant because a negative number of sodas or pizzas does not make sense in this context.
Finally, we will use a test point to know which region should be shaded. For simplicity, we will substitute ( 0, 0) into our inequality.
We can see that substituting the point resulted in a true statement. Therefore, we will shade the region below the boundary line.
This corresponds to option A.