Representing Inequalities on a Number Line
Inequalities compare two expressions, indicating that one is less than, greater than, or equal to the other. When representing inequalities involving one variable on a number line, it is important to show whether the boundary value is included in the solution set.
For strict inequalities such as x > 3 or x < 5, the boundary point (3 or 5) is not included. We represent this by drawing an open circle at the boundary point and shading the number line in the direction of the inequality.
For inclusive inequalities such as x ≥ 2 or x ≤ 7, the boundary point is included. We represent this by drawing a closed (filled) circle at the boundary point and shading accordingly. This visual distinction helps clarify which values satisfy the inequality.
- Open circle for strict inequalities (<, >) indicating the boundary is not included.
- Closed circle for inclusive inequalities (≤, ≥) indicating the boundary is included.
- Shade the number line to the right for greater than inequalities and to the left for less than inequalities.
Solving Simple Linear Inequalities
To solve linear inequalities such as 3x < 2x + 4 or 3x - 2 < 7, the goal is to isolate the variable on one side to find the range of values that satisfy the inequality.
For example, solve 3x < 2x + 4: subtract 2x from both sides to get 3x - 2x < 4, which simplifies to x < 4. This means all values of x less than 4 satisfy the inequality.
Similarly, solve 3x - 2 < 7: add 2 to both sides to get 3x < 9, then divide both sides by 3 to get x < 3. The solution set is all x less than 3.
- Rearrange terms to isolate the variable.
- Perform the same operation on both sides, remembering to reverse the inequality sign when multiplying or dividing by a negative number.
- Express the solution as an inequality or on a number line with appropriate open or closed circles.
Graphical Representation of Linear Inequalities in Two Variables
When inequalities involve two variables, such as y > 2x + 1 or y ≤ -x + 3, their solutions form regions on the coordinate plane.
First, draw the boundary line given by the corresponding equation (e.g., y = 2x + 1). Use a broken (dashed) line if the inequality is strict (< or >), indicating points on the line are not included. Use a solid line if the inequality is inclusive (≤ or ≥), indicating points on the line are included in the solution.
Next, determine which side of the boundary line satisfies the inequality by testing a point not on the line, commonly (0,0). If the test point satisfies the inequality, shade the region containing that point; otherwise, shade the opposite side. The shaded region represents all solutions to the inequality.
- Draw boundary line: broken for strict inequalities, solid for inclusive.
- Test a point to decide which side to shade.
- Shade the solution region; leave the other side unshaded.
Interpreting and Combining Inequalities to Define Regions
Multiple inequalities can be combined to define a specific region on the coordinate plane. For example, the region satisfying y ≥ x and y ≤ 2x + 3 is the area where the shaded regions of both inequalities overlap.
To list inequalities defining a given region, identify the boundary lines and determine whether each boundary is included (solid line) or excluded (broken line). Then write the inequalities corresponding to each boundary, ensuring the correct inequality direction to include the region.
This process helps in understanding constraints and feasible regions in problems, excluding more advanced linear programming contexts.
- Identify boundary lines and their types (solid or broken).
- Determine inequality directions based on shading.
- Write inequalities that together define the region.
At-a-glance summary
Comparison table
| Step | Detail | Example | Exam Use |
|---|---|---|---|
| Representing inequalities on number lines | Use open circles for < or >, closed circles for ≤ or ≥, shade accordingly | x > 3: open circle at 3, shade right | Show correct circle type and shading for full marks |
| Solving linear inequalities | Isolate variable, perform operations carefully, reverse inequality if multiplying/dividing by negative | 3x - 2 < 7 → x < 3 | Show all steps and check solution |
| Graphing inequalities in two variables | Draw boundary line (broken for strict, solid for inclusive), test point to decide shading | y ≤ 2x + 1: solid line y=2x+1, test (0,0), shade below | Draw correct line type and shade correct region |
| Listing inequalities for a region | Identify boundaries and inclusion, write inequalities with correct directions | Region bounded by y ≥ x and y < 3x - 2 | List all inequalities defining the region accurately |
Active recall
Quick check
1. How do you represent x ≥ 5 on a number line?
Draw a closed circle at 5 and shade the number line to the right (values greater than or equal to 5).
2. Solve the inequality 4x - 3 < 9.
Add 3 to both sides: 4x < 12; divide both sides by 4: x < 3. Solution: all x less than 3.
3. What type of line do you draw for y > 2x - 1 on a graph?
Draw a broken (dashed) line for y = 2x - 1 because the inequality is strict (>).
4. How do you decide which side of the boundary line to shade for y ≤ x + 2?
Test a point not on the line, such as (0,0). Substitute: 0 ≤ 0 + 2 → 0 ≤ 2 (true), so shade the side containing (0,0).
5. List inequalities defining a region bounded by y ≥ 0 and x ≤ 4.
The region is defined by y ≥ 0 (above or on the x-axis) and x ≤ 4 (left of or on the vertical line x=4).