Linear Inequalities

The One Thing

Linear inequalities are solved by applying operations that preserve the order relationship, reversing the inequality sign only when multiplying or dividing by a negative number. Their solutions are sets of values, intervals, or shaded half-planes rather than usually a single value.

Definitions and Results

• Linear inequality: A statement comparing two linear expressions, such as $2x+3<9$ or $x+y\le 5$. The symbols $<$ and $>$ represent strict inequalities; $\le$ and $\ge$ represent inclusive inequalities.
• Solution of an inequality: Any value or ordered pair that makes the inequality true.
• Equivalent inequality: An inequality having exactly the same solution set as the original inequality.
• One-variable inequality: An inequality involving one variable, such as $3x-2\ge 7$.
• Two-variable inequality: An inequality involving two variables, such as $2x+y<6$.
• Boundary line: The line obtained by replacing the inequality sign with an equality sign. It separates the coordinate plane into regions.
• Half-plane: One of the two regions into which a boundary line divides the plane.
• Test point: A point selected from a region to determine whether that entire region satisfies the inequality.
• Interval notation: A compact representation of solution sets. Brackets indicate included endpoints, while parentheses indicate excluded endpoints.
• Preservation of order: Adding or subtracting the same number on both sides does not change the inequality direction. Multiplying or dividing by a positive number also leaves the sign unchanged.
• Reversal of order: Multiplying or dividing both sides by a negative number reverses the inequality sign.
• General one-variable result: For $ax+b<c$:

- if $a>0$, then $x<\frac{c-b}{a}$; - if $a<0$, then $x>\frac{c-b}{a}$.

• Number-line conventions: Use an open circle for $<$ or $>$, and a closed circle for $\le$ or $\ge$.
• Two-variable graph: To graph $ax+by<c$ or $ax+by>c$, first draw $ax+by=c$. Use a dashed boundary line for $<$ or $>$, and a solid boundary line for $\le$ or $\ge$.
• Test-point principle: A convenient test point is often $(0,0)$, provided it does not lie on the boundary line. Testing one point in a region is sufficient because a linear expression has the same inequality status throughout each open half-plane.
• System of linear inequalities: The solution is the common region, or intersection, satisfying every inequality.
• Applied restrictions: Variables must be defined clearly. Restrictions such as $x\ge 0$ may be required when variables represent counts, distances, or prices.

Worked Methods

Solving a one-variable inequality

• Simplify the inequality using addition or subtraction on both sides.
• Isolate the variable using multiplication or division.
• If multiplying or dividing by a negative number, reverse the inequality sign.
• Express the result algebraically, on a number line, or in interval notation.

Example: $2x-3<7$

$$2x-3<7$$

Add $3$ to both sides:

$$2x<10$$

Divide by $2$, which is positive:

$$x<5$$

The solution is $x<5$, represented by an open circle at $5$ with shading to the left.

Example: $-3x+6\ge 0$

$$-3x+6\ge 0$$

Subtract $6$ from both sides:

$$-3x\ge -6$$

Divide by $-3$ and reverse the sign:

$$x\le 2$$

The solution is $x\le 2$, represented by a closed circle at $2$ with shading to the left.

Graphing a two-variable inequality

• Replace the inequality sign with an equality sign to obtain the boundary line.
• Draw the boundary line.
• Use a dashed line for a strict inequality, $<$ or $>$.
• Use a solid line for an inclusive inequality, $\le$ or $\ge$.
• Select a test point, often $(0,0)$ if it is not on the boundary.
• Substitute the test point into the original inequality.
• Shade the half-plane containing the points that satisfy the inequality.

For an inequality of the form $ax+by<c$ or $ax+by>c$, the boundary is $ax+by=c$, while the original inequality determines which side is shaded.

Solving a system of linear inequalities

• Graph the boundary line for each inequality.
• Apply the appropriate dashed or solid convention to each boundary.
• Identify the half-plane satisfying each inequality.
• Shade or identify the region common to all the individual solution regions.
• State the intersection as the solution set.

Where It Goes Wrong

• Dividing or multiplying by a negative number without reversing the inequality sign, as in $-3x\ge -6$, which gives $x\le 2$, not $x\ge 2$.
• Using an open circle or dashed boundary for an inclusive inequality, or a closed circle or solid boundary for a strict inequality.
• Forgetting that the equality form determines the boundary line, while the original inequality determines the shaded side.
• Choosing $(0,0)$ as a test point when it lies on the boundary line.
• Giving only a single endpoint instead of the complete solution set, which may need interval notation, a number-line graph, or a shaded half-plane.
• Omitting restrictions such as $x\ge 0$ when variables represent counts, distances, or prices.

What Gets Asked

• Define a linear inequality, solution, equivalent inequality, boundary line, half-plane, test point, or interval notation.
• Solve a one-variable inequality and represent the result on a number line or in interval notation.
• Explain why the inequality sign reverses when dividing or multiplying by a negative number.
• Solve examples such as $2x-3<7$ and $-3x+6\ge 0$.
• Graph a two-variable inequality by drawing its boundary line, selecting a test point, and shading the correct half-plane.
• Determine the solution region for a system of linear inequalities by finding the intersection of the individual regions.
• Translate an applied situation into an inequality, defining variables and including restrictions such as $x\ge 0$.