Solve Inequalities With Absolute Value - Tutorials


Solve Inequalities With Absolute Value - Tutorials

This is a tutorial on solving inequalities with absolute value. Detailed solutions and explanations are provided.

Example 1: Solve the inequality.

\( |x + 2| \lt 3 \)

Solution to example 1

  • The above inequality is solved by writing a double inequality equivalent to the given inequality but without absolute value

    \(- 3 \lt x + 2 \lt 3 \)

  • Solve the double inequality to obtain

    \(- 5 \lt x \lt 1 \)

  • The above solution set is written in interval form as follows

    \((-5 , 1)\)



Example 2: Solve the inequality.

\(|- 2 x - 4| \gt 9\)

Solution to example 2

  • Solving the above inequality is equivalent to solving

    \(-2 x - 4 \gt 9\) or \(-2 x - 4 \lt - 9\)

  • Which gives

    \( x \lt -13 / 2\) or \( x \gt 5 / 2 \)

  • The above solution set is written in interval form as follows

    \( (-\infty , -13 / 2) \cup (5 / 2 , + \infty) \)



Example 3: Solve the inequality.

\( x + 2 \lt |x^2 - 4| \)

Solution to example 3

  • Condition 1 - For \( x^2 - 4 \gt = 0 \), or \( x\) in the interval \( (-\infty , -2] \cup [2 , +\infty)\), we can write

    \( |x^2 - 4| = x^2 - 4 \)

  • Substitute the expression with the absolute value in the given inequality and solve

    \( x + 2 \lt x^2 - 4\)

    \( x^2 - x - 6 \gt 0\)

  • The solution set to above inequality is given by the interval
    \( (-\infty , -2) \cup (3 , +\infty)\)

  • The intersection of intervals \((-\infty , -2] \cup [2 , +\infty)\) and \((-\infty , -2) \cup (3 , +\infty)\) gives the solution set

    \((-\infty , -2) \cup (3 , +\infty)\)

    Condition 2 - For \(x^2 - 4 \lt 0\), or \(x\) in the interval \((-2 , 2)\), we can write

    \( |x^2 - 4| = -(x^2 - 4) \)

  • Substitute the expression with the absolute value in the given inequality and solve

    \(x + 2 \lt -(x^2 - 4)\)

    \(x^2 + x - 2 \lt 0\)

  • The solution set to above inequatlity is given by the intersection of the intervals \((-2 , 1)\) and \((-2 , 2)\) which gives

    (-2 , 1)

  • Conclusion: The solution set to the given inequality is \((-\infty, -2) \cup (-2 , 1) \cup (3 , +\infty)\)

Check the above answer to the inequality graphically.(see graph below).

graph of x + 2 and |x^2 - 4| to check answer to inequality above.

More references and links on how to Solve Equations, Systems of Equations and Inequalities.

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