Solving Linear Inequalities in One and Two Variables

Symbols and Labelling Conventions

Inequalities are represented by four primary symbols. For a single variable $x$, the representation on a number line follows specific conventions to indicate whether the boundary value is included in the solution set.

1. $<$ (Less than): $x < 4$ defines all values less than but not including 4. On a number line, this is shown with an open circle at 4.
2. $>$ (Greater than): $x > 4$ defines all values greater than but not including 4. This is also shown with an open circle at 4.
3. $\leq$ (Less than or equal to): $x \leq 4$ defines all values less than 4, including 4 itself. This is shown with a solid circle at 4.
4. $\geq$ (Greater than or equal to): $x \geq 4$ defines all values greater than 4, including 4. This is shown with a solid circle at 4.

Inequalities in Two Variables

When dealing with two variables, solution sets are represented as shaded regions on a coordinate graph. The boundary is defined by the equivalent linear equation.

• Dotted Lines: Used for strict inequalities ($<$ or $>$). This indicates that the points on the line itself are not part of the solution.
• Solid Lines: Used for inclusive inequalities ($\leq$ or $\geq$). This indicates that the points on the line are included.

The shaded area above shows the region $x < y$. Notice the dotted line for the boundary $x = y$.

In this graph, the region $x \geq y$ is shown with a solid boundary line. When solving problems involving multiple inequalities, it is often clearer to shade the unwanted region and leave the required region unshaded. It is crucial to state clearly which method you are using.

Simplifying Inequalities

If $x < y$, specific rules apply when manipulating the expression:

• Addition and Subtraction: Adding or subtracting a real number from both sides leaves the sign unchanged. For example: $x - 4 < y - 4$.
• Positive Multipliers: Multiplying or dividing by a positive real number leaves the sign unchanged. For example: $\frac{3}{4}x < \frac{3}{4}y$.
• Inversion: If $x$ and $y$ are both positive or both negative, inverting both sides changes the direction of the sign. For example: $\frac{1}{x} > \frac{1}{y}$.
• Negative Multipliers: Multiplying or dividing by a negative real number changes the direction of the sign. For example: $-\frac{3}{4}x > -\frac{3}{4}y$.

Combining Inequalities

Combined inequalities take the form $a < b < c$. Both symbols must point in the same direction. Notation such as $a < b > c$ is mathematically incorrect.

• The set $x > -7$ and $x \leq 10$ is written as $-7 < x \leq 10$.
• The set $x < -7$ and $x \leq 10$ simply reduces to $x < -7$.
• If inequalities do not overlap or define a continuous range, such as $x < -7$ and $x \geq 10$, they cannot be combined into a single continuous statement.
• Statements like $-7 < x < -10$ are incorrect because $-7$ is not less than $-10$.

Worked Examples: Single Variable

Example 1: Simple Linear Inequality Solve $3x - 4 > 8$ and represent it on a number line.

1. Add 4 to both sides: $3x > 12$.
2. Divide by 3: $x > 4$. This is represented by a number line with an open circle at 4 and an arrow pointing to the right.

Example 2: Two-sided Linear Inequality Solve $20 \geq 3x - 4 > 8$.

1. Add 4 to all three sections: $24 \geq 3x > 12$.
2. Divide all three sections by 3: $8 \geq x > 4$. On a number line, place an unfilled circle at 4 and a filled circle at 8, joining them with a line.

Example 3: Complicated Variable Expression Solve $3x - 5 < 2(2 + 3x)$.

1. Multiply out the bracket: $3x - 5 < 4 + 6x$.
2. Subtract $6x$ from both sides: $-3x - 5 < 4$.
3. Add 5 to both sides: $-3x < 9$.
4. Multiply by $-1$ and reverse the sign: $3x > -9$.
5. Divide by 3: $x > -3$.

Worked Examples: Finding Ranges

Overlapping Ranges: Find the range for $3x - 4 < 8$ and $3x - 2 > -8$.

• From the first: $3x < 12 \rightarrow x < 4$.
• From the second: $3x > -6 \rightarrow x > -2$.
• Combined: $-2 < x < 4$.

Contained Ranges: Find the range for $3x - 4 < 8$ and $3x - 2 < -8$.

• From the first: $x < 4$.
• From the second: $x < -2$.
• The valid range for both is simply $x < -2$.

Non-overlapping Ranges: Find the range for $3x + 2 < -10$ and $3x - 4 > 8$.

• From the first: $x < -4$.
• From the second: $x > 4$.
• There is no range where both are valid.

Simultaneous Inequalities in Two Variables

To solve $3x + 4y \leq 12$ graphically:

1. Draw the boundary line $3x + 4y = 12$.
2. Test a point, such as $(0,0)$. Since $3(0) + 4(0) \leq 12$ is true, the region containing $(0,0)$ is the solution set.
3. Use a solid line for the boundary.

To solve multiple inequalities like $3x + 4y \leq 12$, $y \geq x + 6$, and $x > -7$:

• Draw each boundary line.
• Shade out the unwanted regions for each constraint.
• The remaining unshaded region is the solution.