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CAT Solving linear and quadratic inequalities - Practice Questions & MCQ

Edited By admin | Updated on Oct 04, 2023 04:20 PM | #CAT

Syllabus

Quick Facts

5 Questions around this concept.

Solve by difficulty

EASY

The solution of the inequality

$\frac{\left ( x+1 \right )\left ( x-1 \right )^{4}}{\left ( x-3 \right )^{2}\left ( x-2 \right )}\geq 0$ is

$\left ( -\infty,-1 \right )\cup \left ( 2,\infty\right )$

$\left ( -\infty ,-1\right ]\cup \left ( 2,\infty \right )$

$\left ( -\infty,-1 \right ] \cup \left ( 2,\infty \right )$ $\cup \{1\}$

none of these

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Concepts Covered - 1

Solving linear and quadratic inequalities

Study Notes: Solving Linear and Quadratic Inequalities

Concept Overview:

Solving linear and quadratic inequalities is an important topic in the study of equations.
These problems involve finding the values of variables that satisfy the given inequality.
Linear inequalities involve equations with variables of degree 1, while quadratic inequalities involve equations with variables of degree 2.

Key Topics:

Linear Inequalities
Quadratic Inequalities

1. Linear Inequalities:

In linear inequalities, the variables have a degree of 1.
They can be solved using techniques like graphing, substitution, and the concept of intervals.
For a linear inequality of the form $\mathrm{ax + b > c}$ , the solution can be found by isolating the variable x on one side of the inequality and determining the sign of the inequality based on the coefficient of x.
For inequalities with two variables, a graph can be used to identify the region that satisfies the inequality.

Example:

Question: Solve the linear inequality $\mathrm{ 3x + 5 > 10. }$

Solution: Subtracting 5 from both sides, we get $\mathrm{3x > 5.}$ Dividing by $\mathrm{3, x > 5/3}$ . The solution to the inequality is $\mathrm{x > 5/3.}$

2. Quadratic Inequalities:

In quadratic inequalities, the variables have a degree of 2.
They can be solved using methods like graphing, factoring, or using the quadratic formula.
The solution to a quadratic inequality may involve multiple intervals or regions that satisfy the inequality.

Example:

Question: Solve the quadratic inequality $\mathrm{x^2 + 2x - 3 > 0. }$

Solution: Factoring the quadratic expression, we get $\mathrm{(x - 1)(x + 3) > 0}$ . To find the solution, we create a sign chart and check the intervals where the expression is positive. The solution is $\mathrm{ x > 1 \: or \: x < -3.}$

Tips and Tricks:

Identify the type of inequality (linear or quadratic) and use the appropriate solving methods.
When solving quadratic inequalities, always factorise the expression and analyse the sign chart.
It's essential to be careful with the signs when multiplying or dividing by negative numbers during solving processes.
Always check the solution by substituting the obtained values back into the original inequality to ensure its validity.
Regular practice of solving inequalities will improve speed and accuracy.

By following the above study notes, understanding the methods, solving previous year questions, and utilising the tips and tricks, students can enhance their problem-solving skills in solving linear and quadratic inequalities for management entrance exams.