Theory Insight:
A compound inequality of the form \(a \leq f(x) \leq b\) represents two simultaneous inequalities:
\[
a \leq f(x) \quad \text{and} \quad f(x) \leq b
\]
The solution is the intersection of both conditions. This is a foundational concept for interval notation and graphical interpretation on the number line.
Solution Roadmap:
Break the compound inequality into two separate inequalities
Solve each inequality step-by-step
Find the common solution (intersection)
Express answer in interval form and visualize on number line
Theory Insight:
When solving compound inequalities involving multiplication or division by a negative number,
the inequality sign reverses. This is a critical rule:
\[
\text{If } a < b \Rightarrow -a > -b
\]
The given inequality represents two simultaneous conditions, and the final solution is their intersection.
Theory Insight:
When inequalities involve fractions, it is often efficient to eliminate denominators first
(by multiplying throughout by the LCM). Also, remember:
Multiplying by a positive number → inequality sign remains same
Multiplying by a negative number → inequality sign reverses
This is a compound inequality, so we solve two parts and take their intersection.
Solution Roadmap:
Split the compound inequality into two inequalities
Remove fraction by multiplying with 2
Solve each inequality carefully
Handle sign reversal correctly when dividing by negative
Intersect both solutions and express in interval form
Theory Insight:
In compound inequalities involving fractions, multiply throughout by the LCM to simplify.
Since we multiply by a positive number (5), the inequality signs remain unchanged.
Always treat strict inequalities (<, >) carefully while writing interval notation.
Solution Roadmap:
Split the compound inequality into two parts
Multiply throughout by 5 to remove denominator
Solve each inequality step-by-step
Combine results and determine correct interval type
Theory Insight:
First simplify the expression carefully. Note:
\[
\frac{3x}{-5} = -\frac{3x}{5}
\Rightarrow 4 - \left(-\frac{3x}{5}\right) = 4 + \frac{3x}{5}
\]
Reducing sign complexity early avoids major algebra mistakes. Then proceed as a standard compound inequality.
Solution Roadmap:
Simplify the expression (handle negative denominator)
Split into two inequalities
Multiply by 5 to eliminate fraction
Solve both and take intersection
Write correct interval using strict/non-strict bounds
Theory Insight:
For compound inequalities involving a fraction, multiply the entire inequality by the denominator
(if positive) to simplify in one step. Since 2 is positive, inequality signs remain unchanged.
This “single-step method” is faster and reduces algebraic errors.
Theory Insight:
When two inequalities are given together, we solve them separately and take their intersection.
Since both are strict inequalities, the final interval will be open at both ends.
Solution Roadmap:
Solve each inequality independently
Isolate \(x\) in both cases
Take intersection of both results
Represent solution on number line using open circles
Theory Insight:
This is a system of linear inequalities. Each inequality gives a range of values of \(x\).
The final solution is the intersection of both ranges. Expansion and simplification must be done carefully.
Solution Roadmap:
Expand both inequalities
Simplify and isolate \(x\)
Find solution of each inequality
Take intersection and express as interval
Solution
$$\begin{aligned}
2(x - 1) &< x + 5 \\
2x - 2 &< x + 5 \\
x &< 7 \\\
3(x + 2) &> 2 - x \\
3x + 6 &> 2 - x \\
4x &> -4 \\
x &> -1 \\\
\therefore \; x \in (-1, 7)
\end{aligned}$$
Why this matters (Boards + Competitive Exams):
Standard system of inequalities — very frequent in CBSE exams
Theory Insight:
In a system of inequalities, the final solution is the intersection of all individual solutions.
Sometimes one inequality gives a stronger restriction than the other — this is called the
dominant constraint.
Solution Roadmap:
Solve each inequality separately
Reduce to simplest form
Compare results and identify the stricter condition
Write final solution as intersection
Solution
$$\begin{aligned}
3x - 7 &> 2(x - 6) \\
3x - 7 &> 2x - 12 \\
x &> -5 \\\
6 - x &> 11 - 2x \\
6 + x &> 11 \\
x &> 5 \\\
\therefore \; x > 5
\end{aligned}$$
Why this matters (Boards + Competitive Exams):
Illustrates concept of dominant inequality (important for faster solving)
Common CBSE pattern where one inequality becomes redundant
Very useful in JEE/NEET for constraint tightening problems
Builds intuition for feasible regions extending to infinity
Theory Insight:
In systems of inequalities, expand expressions carefully and reduce to linear form.
Pay special attention when dividing by negative numbers — inequality signs must reverse.
Final solution is the intersection of both inequalities.
A solution is to be kept between 68°F and 77°F. What is the range in temperature
in degree Celsius (C) if \(F = \dfrac{9}{5}C + 32\)?
Theory Insight:
Temperature conversion between Fahrenheit and Celsius follows a linear relation.
When a range is given in one unit, convert it into the other by forming a compound inequality.
Since the transformation is linear and increasing, the order of inequality remains unchanged.
A solution of 8% boric acid is to be diluted by adding a 2% boric acid solution.
The resulting mixture must be more than 4% but less than 6%.
If 640 litres of 8% solution is given, how many litres of 2% solution should be added?
Theory Insight:
In mixture problems, concentration is defined as:
\[
\text{Concentration} = \frac{\text{amount of solute}}{\text{total volume}}
\]
The final concentration must lie between two values, forming a compound inequality.
This is a classic weighted-average inequality problem.
How many litres of water must be added to 1125 litres of a 45% acid solution so that the
resulting mixture contains more than 25% but less than 30% acid?
Theory Insight:
When water is added, the amount of solute (acid) remains constant, while total volume increases.
Hence, concentration decreases. Such problems are modelled using:
\[
\text{Concentration} = \frac{\text{fixed solute}}{\text{new volume}}
\]
This leads to a compound inequality.
IQ of a person is given by \(IQ=\dfrac{MA}{CA}\times 100\).
If \(80 \leq IQ \leq 140\) for a group of 12-year-old children, find the range of their mental age.
Theory Insight:
This is a direct application of linear transformation in inequalities.
Since IQ is proportional to mental age (MA), the inequality can be solved by simple scaling.
As all multipliers are positive, inequality signs remain unchanged.
