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Note To follow this tutorial, you should know how to
graph linear equations and solve systems of two equations in two unknowns.
Inequalities
We start by summarizing some facts about inequalities:
Warmup: Inequalities
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Strict inequalities
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| Notation | Meaning | Examples |
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$a \lt b$ | $a$ is less than $b.$ | $4 \lt 6, \ \ -1 \lt 0$ | $a \gt b$ | $a$ is greater than $b.$ | $6 \gt 4, \ \ 0 \gt -1$ |
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Non-strict inequalities
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| Notation | Meaning | Examples |
|---|
$a \leq b$ | $a$ is less than or equal to $b.$ | $-1 \leq 2, \ \ 2 \leq 2$ | $a \geq b$ | $a$ is greater than or equal to $b.$ | $2 \geq -1, \ \ 2 \geq 2$ |
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Some examples for you
Manipulating inequalities
The following rules are stated using $\leq$, but also apply to all the other inequalities.
| Rule | Formula | Example |
|---|
Add or subtract a quantity from both sides: | $x \leq y \Rightarrow x \pm a \leq y \pm a$ | $x \leq 2y \Rightarrow x - 2y \leq 0$ | Multiply or divide both sides by a positive constant. | $x \lt y $ and $a > 0 \Rightarrow ax \leq ay$ | $x \lt y \Rightarrow \frac{x}{4} \leq \frac{y}{4} $ | Multiply or divide both sides by a negative constant and reverse the inequality. | $x \lt y $ and $a < 0 \Rightarrow ax \geq ay$ | $x \lt y \Rightarrow -4x \geq -4y $ | Swap left- and right-hand sides, and reverse the inequality. | $x \lt y \Rightarrow y \geq x$ | $x \lt 2y \Rightarrow 2y \geq x $ |
Some examples for you
Note For the rest of this tutorial, we will be discussing only non-strict inequalities: $\leq$ and $\geq$, and of a particular type.
Linear inequalities in two unknowns
The kinds of inequalities we are interested in here are the
linear (non-strict) inequalities, specifically those with two unknowns $x$ and $y$:
Linear inequalities in x and y
A
linear inequality in $x$ and $y$ has the form
$ax + by \leq c \quad$ or $\quad ax + by \geq c \qquad$
Examples
| $3x - y \leq 5$ | | $a = 3, b = -1, c = 5$ |
| $y \leq 50$ | | $a = 0, b = 1, c = 50$ |
| $x \geq 0$ | | $a = 1, b = 0, c = 0$ |
Also,
$2y \leq 3x$ can be rewritten as $-3x + 2y \leq 0$. ($a = -3, b = 2, c = 0$)
Solutions of linear inequalities in x and y
A
solution of an inequality in $x$ and $y$ consists of a pair of numbers $(x, y)$: a value for $x$ and a value for $y$ that satisfy the inequality.
How to sketch the solution set of a linear inequality ax + by ≤ c or ax + by ≥ c
- Sketch the line of the associated equation $ax = by = c.$. This is the bounding line of the solution set.
- Choose a "test point" $(x, y)$ not on the line you sketched and check whether its coordinates satisfy the inequality.
- If they do, then the solution set consists of the line plus the entire region on the same side of the test point.
If not, then the solution set consists of the line plus the entire region on the other side of the test point.
- Grey out the rest of $xy$-plane (do not shade the solution set), so the solution set is the portion left white.
Examples
The solutions of $3x + 2y \leq 4$ inlcude all the solutions of the associated
equation $3x + 2y = 4$ (represented by points on the corresponding line, such as $(2, -1)$) as well as others that represent points
below the line, such as $(0, 0)$ because they also satisfy the inequality:
$3(2) + 2(-1) \leq 4$ \t ✓
\\ $3(0) + 2(0) \leq 4$ \t ✓
Some examples for you
Decide which of the following are solutions of $3x + 2y \leq 4$
Sketching the solution set
Note We have left the solution set white by "greying-out" the rest of the plane. This is the technique used here and also in the textbook, and it greatly simplifies the work in solving simultaneous inequalities as we will see below.
Consider the following inequality:
First, obtain the boundary line using the graph below (Adjust the lines by dragging the points marked with "x" and press "Check".)
Solving simultaneous inequalities
The solution set of a system or two or more inequalities in $x$ and $y$ is the set of points $(x, y)$ that satisfy
all the inequalities.
How to sketch the solution set of a system- Determine the solution set of the first inequality.
- Grey out the rest of $xy$-plane (do not shade the solution set).
- Repeat (1) and (2) for each subsequent inequality.
- When done, the portion of the plane left white, plus it boundary, is the solution set.
Examples
Consider the system
$3x + 2y \leq 4$
$x - 2y \leq 0$
Steps (1) and (2) for $3x + 2y \leq 4$
Solution set
for $3x + 2y \leq 4$
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Steps (1) and (2) for $x - 2y \leq 0$
Solution set
for $x - 2y \leq 0$
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Result:
Solution set
for the system |
Consider the following system:
First, obtain the boundary line for the
first inequality using the graph below (Adjust the lines by dragging the points marked with "x" and press "Check".)
Application: Setting up a system of linear inequalities
The
Scrumptious %30 company makes two brands of %30 for vendors: %31, using %51 of %34 and %53 of %35 per %33, and %32, using %52 of %34 and %54 of %35 per %33. Each day the company uses a total of %38 %55 %36s of %34 and %39 %56 %36s of %35. The company bylaws stipulate that it make %40 %41 times as much %42 as %43. Let $x$ be the number of %37 of %31, and let $y$ be the number of %37 of %32.
Express these constraints as a system of linear inequalities in $x$ and $y:$
Now try the exercises in %4, some the %8, or move ahead to the next tutorial by pressing "Next tutorial" on the sidebar.
Last Updated: August, 2016
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