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Here are a few recommended readings before getting started with this lesson.
When graphing a linear equation, the resulting graph is a line.
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.
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.
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.
x=1, y=1
Identity Property of Multiplication
Add terms
Consider the following graph of a linear inequality represented on a coordinate plane.
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.
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.