Solve the inequality and graph the solution on the real number line. Use a graphing utility to v

step 1 According to the question we have given equation is x plus 12 minus 3 x plus 2 upon x plus 2 is

Solve the inequality and graph the solution on the real...

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Key Concepts

Rational Inequalities

A rational inequality involves a ratio of two polynomials. Solving these inequalities often requires transforming the expression so that one side is zero, then analyzing when the resulting rational function is positive or negative.

Domain Restrictions

When working with rational expressions, it is essential to identify values that cause the denominator to be zero, as these are not included in the domain. This step ensures that the solution only includes values where the expression is defined.

Critical Points

Critical points are the values that make either the numerator or the denominator zero. These points divide the number line into intervals and are key in determining where the inequality changes its sign.

Test Intervals

After determining the critical points, the number line is split into intervals. Each interval is then tested with a representative value to establish whether the original inequality holds, ensuring that the correct sections of the number line are included in the final solution.

Graphical Representation of Solutions

Graphing the solution on the real number line, and using graphing utilities, provides a visual confirmation of the analytical solution. This process helps verify which intervals satisfy the inequality and accurately represents boundary conditions and discontinuities.

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