Edexcel IAL - Pure Maths 1- 1.7 Graphical Interpretation of Linear and Quadratic Inequalities- Study notes - New syllabus
Edexcel IAL – Pure Maths 1- 1.7 Graphical Interpretation of Linear and Quadratic Inequalities -Study notes- New syllabus
Edexcel IAL – Pure Maths 1- 1.7 Graphical Interpretation of Linear and Quadratic Inequalities -Study notes -Edexcel A level Physics – per latest Syllabus.
Key Concepts:
- 1.7 Graphical Interpretation of Linear and Quadratic Inequalities
Linear Inequalities Interpreted Graphically
A linear inequality compares two expressions and defines a region on the number line or coordinate plane where the inequality holds. When represented graphically, the inequality corresponds to the region on one side of a straight line.
Consider the linear inequality:
\( ax + b > cx + d \)
This can be rearranged to:
\( (a – c)x > (d – b) \)
Graphically, the solution set corresponds to all \( x \)-values for which the point on the line \( y = ax + b \) lies above or below the line \( y = cx + d \), depending on the inequality symbol.
Key Concepts for Linear Inequalities (Graphical Interpretation)
| Concept | Explanation |
| Inequality between lines | Comparing \( ax + b \) and \( cx + d \) means comparing the heights of two lines. If \( ax + b > cx + d \), then the graph of \( y = ax + b \) lies above the graph of \( y = cx + d \). |
| Solution on number line | The solution set is a range of \( x \)-values where the inequality is true. |
| Strict vs non-strict | \( > \) or \( < \): boundary line is not included \( \ge \) or \( \le \): boundary line is included |
| Graphical interpretation of expressions | Inequalities such as \( px^2 + qx + r > 0 \), \( px^2 + qx + r < ax + b \) are interpreted by comparing a quadratic curve with a straight line. |
| Fractions and brackets | Inequalities with fractions/brackets can be rearranged to linear or quadratic form. Example: \( \dfrac{a}{x} < b \) becomes \( a < bx \), with the extra condition \( x \neq 0 \), and sign reversal if multiplying/dividing by a negative \( x \). |
Understanding Linear Inequalities Graphically
- Plot the line corresponding to the boundary equality (e.g., \( ax + b = cx + d \)).
- Determine which side of the line satisfies the inequality.
- For inequalities involving functions (e.g., a curve below a line), compare vertical heights.
- For fractional inequalities, note sign changes depending on the value of \( x \).
Example
Solve the inequality \( 3x + 2 > x + 8 \) and interpret it graphically.
▶️ Answer / Explanation
\( 3x + 2 > x + 8 \Rightarrow 2x > 6 \Rightarrow x > 3 \)
Graphical meaning:
The line \( y = 3x + 2 \) lies above the line \( y = x + 8 \) whenever \( x > 3 \).
Example
Interpret graphically the inequality \( 2x + 1 \le 5 – x \).
▶️ Answer / Explanation
Rearrange:
\( 2x + 1 \le 5 – x \Rightarrow 3x \le 4 \Rightarrow x \le \dfrac{4}{3} \)
Graphical interpretation:
The line \( y = 2x + 1 \) is below the line \( y = 5 – x \) for \( x \le \dfrac{4}{3} \).
Example
Solve and interpret the inequality \( \dfrac{6}{x} < 3 \).
▶️ Answer / Explanation
Case distinction is required because multiplying by \( x \) may reverse the inequality.
Case 1: \( x > 0 \)
\( \dfrac{6}{x} < 3 \Rightarrow 6 < 3x \Rightarrow x > 2 \)
Case 2: \( x < 0 \)
Multiplying by negative \( x \) reverses inequality:
\( 6 > 3x \Rightarrow x < 2 \)
But since \( x < 0 \), this gives all negative \( x \).
Final solution:
\( x > 2 \) or \( x < 0 \), with \( x \neq 0 \).
Graphical interpretation:
Plot \( y = \dfrac{6}{x} \) and compare with the line \( y = 3 \). The inequality holds where the curve lies below the horizontal line.
Quadratic Inequalities Interpreted Graphically
A quadratic inequality involves comparing a quadratic expression with zero or with another function (usually a straight line). Because the graph of a quadratic function is a parabola, the inequality describes the parts of the \( x \)-axis for which the quadratic is above or below the axis or another line.
General Quadratic Inequality
\( px^2 + qx + r \gtrless 0 \)
or
\( px^2 + qx + r \gtrless ax + b \)
Graphically, this compares the curve \( y = px^2 + qx + r \) with \( y = 0 \) or \( y = ax + b \).
Key Concepts for Quadratic Inequalities (Graphical Interpretation)
| Concept | Explanation |
| Compare with 0 | Solve where the parabola is above the x-axis (\( >0 \)) or below it (\(<0\)). |
| Compare with a line | Solve where \( px^2 + qx + r \gtrless ax + b \) by comparing the heights of the curve and the line. |
| Rewrite as single quadratic | Move all terms to one side: \( px^2 + qx + r – ax – b \gtrless 0 \) which simplifies to \( px^2 + (q – a)x + (r – b) \gtrless 0 \). |
| Use discriminant | The discriminant \( \Delta = b^2 – 4ac \) indicates how many x-intercepts the parabola has, which affects intervals of sign. |
| Fractional or bracket inequalities | Inequalities involving fractions can produce quadratic inequalities when rearranged. Example: \( \dfrac{a}{x} < b \) becomes \( a < bx \), and solutions depend on signs of \( x \). |
Method (Graphical Interpretation)
- Rewrite the inequality as a single quadratic on one side.
- Find roots by solving the corresponding quadratic equation.
- Sketch the parabola roughly (shape depends on sign of \( p \)).
- Use the graph to determine where the curve is above or below the line/x-axis.
- Write the solution intervals.
Example
Solve the inequality \( x^2 – 4x + 3 > 0 \) using a graphical interpretation.
▶️ Answer / Explanation
Factorise:
\( x^2 – 4x + 3 = (x – 1)(x – 3) \)
Roots: \( x = 1,\ 3 \)
The parabola opens upward (\( a = 1 > 0 \)).
For \( > 0 \), the curve is above the x-axis:
\( x < 1 \) or \( x > 3 \)
Example
Solve \( x^2 + x – 6 \le 2x + 1 \) using graphical reasoning.
▶️ Answer / Explanation
Bring all terms to one side:
\( x^2 + x – 6 – 2x – 1 \le 0 \)
\( x^2 – x – 7 \le 0 \)
Find roots:
\( x = \dfrac{1 \pm \sqrt{1 + 28}}{2} = \dfrac{1 \pm \sqrt{29}}{2} \)
Because the parabola opens upward, it is below or touching the x-axis between the roots.
Solution:
\( \dfrac{1 – \sqrt{29}}{2} \le x \le \dfrac{1 + \sqrt{29}}{2} \)
Graph interpretation: The curve \( y = x^2 + x – 6 \) lies below the line \( y = 2x + 1 \) between these two intersection points.
Example
Solve the inequality \( \dfrac{3x + 1}{x – 2} > 1 \), noting that it reduces to a quadratic inequality.
▶️ Answer / Explanation
Subtract 1:
\( \dfrac{3x + 1}{x – 2} – 1 > 0 \)
Combine into a single fraction:
\( \dfrac{3x + 1 – (x – 2)}{x – 2} > 0 \)
\( \dfrac{2x + 3}{x – 2} > 0 \)
Critical values: \( x = -\dfrac{3}{2} \) (numerator zero), \( x = 2 \) (denominator zero → undefined).
Sign analysis gives intervals:
- Both numerator and denominator positive → \( x > 2 \)
- Both negative → \( -\dfrac{3}{2} < x < 2 \)
Solution:
\( -\dfrac{3}{2} < x < 2 \quad \text{or} \quad x > 2 \)
Graph interpretation: The rational graph intersects the line \( y = 1 \) at \( x = -\dfrac{3}{2} \), and there is a vertical asymptote at \( x = 2 \). The inequality holds where the curve lies above the line.