As we have seen in our work with inequalities (see these entries to remind yourself if you need), an inequality states a range of solutions to a problem instead of just a singular answer.
The difference is best described with an example:
Here is the line $y=2x+3$y=2x+3
The line shows all the solutions to the equation. All the possible $y$y values that make this equation true for any $x$x value that is chosen.
For every $x$x value there is only one possible corresponding $y$y value.
For example, if $x=1$x=1, then according to the equation $y=5$y=5 (as marked on the diagram)
Here is the inequality $y>2x+3$y>2x+3
The solution to this is not a single line, as for every $x$x value, there are multiple $y$y values that satisfy the inequality. The solution graph is therefore a region.
A coloured in space indicating all the possible coordinates $\left(x,y\right)$(x,y) that satisfy the inequality.
For example, at $x=1$x=1, $y>5$y>5. So any coordinate with an $x$x value of $1$1 and a $y$y value larger than $5$5 is a solution.
The dotted line corresponds to the strictly greater than symbol that was used. That is, since $y$y cannot equal $2x+3$2x+3, we cannot include the points on the line.
Here is another example $y\le2x+3$y≤2x+3
Again we have a region, and this time we also have solid line indicating that the $y$y value can be less than or EQUAL to $2x+3$2x+3, for any given $x$x.
For example, if we choose $x=3$x=3, the points that satisfy the inequality are all the points whose $y$y value is less than or equal to $2\times3+3$2×3+3 or $9$9.
There are many points that do this. One such point would be $\left(3,8\right)$(3,8).
Examples
Question 1
Select the inequalities that describe the shaded region.
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A coordinate plane, with the x-axis from $-10$−10 to $10$10 and the y-axis also from $-10$−10 to $10$10. Two lines are drawn on the plane: a $solid$solid horizontal line which crosses the y-axis at $\left(0,-3\right)$(0,−3), and a $solid$solid line which crosses the x-axis at $\left(-\frac{5}{4},0\right)$(−54,0) and the y-axis at $\left(0,-5\right)$(0,−5). These lines intersect, dividing the coordinate plane into four regions. The $\text{upper left}$upper left region is shaded.
$y$y$\ge$≥$-4x-5$−4x−5 or $y$y$\ge$≥$-3$−3
A
$y$y$\ge$≥$-4x-5$−4x−5 and $y$y$\le$≤$-3$−3
B
$y$y$\le$≤$-4x-5$−4x−5 and $y$y$\ge$≥$-3$−3
C
$y$y$\le$≤$-4x-5$−4x−5 and $-\frac{5}{4}$−54$\le$≤$-3$−3
D
Question 2
Select the inequalities that describe the shaded region.
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A graph of lines of $y=x$y=x and $y=-x$y=−x on a cartesian plane is plotted. The line representing $y=x$y=x is illustrated as a dashed line. The line for $y=-x$y=−x is shown as a solid line. At the intersection of the two lines, the area to the left is shaded gray.
$y$y$>$>$x$x and $y$y$\le$≤$-x$−x
A
$y$y$\le$≤$x$x and $y$y$>$>$-x$−x
B
$y$y$>$>$x$xor$y$y$\ge$≥$-x$−x
C
$y$y$<$<$x$xand$y$y$\le$≤$-x$−x
D
Question 3
Outcomes
11.A.LE.1
Linear inequalities. Algebraic solutions of linear inequalities in one variable and their representation on the number line. Graphical solution of linear inequalities in two variables. Solution of system of linear inequalities in two variables – graphically.
