Solve The Following Inequality Which Graph Shows The Correct Solution

Solve the Following Inequality: Which Graph Shows the Correct Solution? A Comprehensive Guide

Solving inequalities is a crucial skill in algebra and forms the basis for understanding many mathematical concepts. This comprehensive guide will walk you through the process of solving inequalities, focusing on interpreting the solution and identifying the correct graphical representation. We'll cover various types of inequalities, including linear, quadratic, and absolute value inequalities, and show you how to represent the solution set on a number line. Understanding this process is essential for success in higher-level mathematics and various applications in science and engineering.

Understanding Inequalities

Before diving into solving inequalities, let's solidify our understanding of the fundamental concepts. An inequality is a mathematical statement that compares two expressions using one of the following inequality symbols:

• <: less than
• >: greater than
• ≤: less than or equal to
• ≥: greater than or equal to

Unlike equations, which have a single solution or a finite set of solutions, inequalities often have an infinite number of solutions. The solution set represents all the values that satisfy the inequality.

Solving Linear Inequalities

Linear inequalities involve a linear expression (an expression where the highest power of the variable is 1). The process of solving them is similar to solving linear equations, with one crucial difference: when multiplying or dividing by a negative number, you must reverse the inequality sign.

Example: Solve the inequality 3x + 5 > 11.

1. Subtract 5 from both sides: 3x > 6
2. Divide both sides by 3: x > 2

The solution to the inequality is x > 2. This means any value of x greater than 2 satisfies the inequality.

Graphical Representation: On a number line, we represent this solution by shading the region to the right of 2. We use an open circle at 2 to indicate that 2 is not included in the solution set.

     <-------------------o------------------->
     -1   0   1   2   3   4   5   6
                   x > 2

Example with Negative Multiplication: Solve the inequality -2x + 4 ≤ 6

1. Subtract 4 from both sides: -2x ≤ 2
2. Divide both sides by -2 (and reverse the inequality sign): x ≥ -1

The solution is x ≥ -1. This means x can be -1 or any value greater than -1.

Graphical Representation: We shade the region to the right of -1, including -1 itself (represented by a closed circle).

     <-------------------•------------------->
     -3  -2  -1   0   1   2   3
                    x ≥ -1

Solving Compound Inequalities

Compound inequalities involve two or more inequalities combined using "and" or "or."

Example (And): Solve the compound inequality -3 < 2x + 1 < 7.

This is equivalent to two separate inequalities: -3 < 2x + 1 AND 2x + 1 < 7. We solve them simultaneously:

1. Subtract 1 from all parts: -4 < 2x < 6
2. Divide all parts by 2: -2 < x < 3

The solution is -2 < x < 3. This means x is greater than -2 and less than 3.

Graphical Representation: Shade the region between -2 and 3, using open circles at -2 and 3.

     <-------------------o---------o------------------->
     -3  -2  -1   0   1   2   3   4
                   -2 < x < 3

Example (Or): Solve the compound inequality x < -1 OR x > 3.

This means x satisfies either x < -1 or x > 3.

Graphical Representation: Shade the region to the left of -1 and the region to the right of 3, using open circles at -1 and 3.

     <-------------------o-------------------o------------------->
     -3  -2  -1   0   1   2   3   4   5
                x < -1  OR  x > 3

Solving Quadratic Inequalities

Quadratic inequalities involve a quadratic expression (an expression where the highest power of the variable is 2). Solving them typically involves factoring the quadratic expression, finding the roots, and testing intervals.

Example: Solve the inequality x² - 4x + 3 < 0.

1. Factor the quadratic: (x - 1)(x - 3) < 0

2. Find the roots: x = 1 and x = 3

3. Test intervals: We test the intervals (-∞, 1), (1, 3), and (3, ∞).

   • (-∞, 1): Choose x = 0. (0 - 1)(0 - 3) = 3 > 0. This interval is not part of the solution.
   • (1, 3): Choose x = 2. (2 - 1)(2 - 3) = -1 < 0. This interval is part of the solution.
   • (3, ∞): Choose x = 4. (4 - 1)(4 - 3) = 3 > 0. This interval is not part of the solution.

The solution is 1 < x < 3.

Graphical Representation: Shade the region between 1 and 3, using open circles at 1 and 3.

     <-------------------o---------o------------------->
     0   1   2   3   4   5
                1 < x < 3

Solving Absolute Value Inequalities

Absolute value inequalities involve the absolute value function, denoted by |x|. Remember that |x| represents the distance of x from 0.

Example: Solve the inequality |x - 2| < 3.

This inequality means the distance between x and 2 is less than 3. This can be rewritten as a compound inequality:

-3 < x - 2 < 3

Solving this compound inequality:

1. Add 2 to all parts: -1 < x < 5

The solution is -1 < x < 5.

Graphical Representation: Shade the region between -1 and 5, using open circles at -1 and 5.

     <-------------------o---------o------------------->
     -2  -1   0   1   2   3   4   5   6
                 -1 < x < 5

Example: Solve the inequality |x + 1| ≥ 2.

This inequality means the distance between x and -1 is greater than or equal to 2. This can be rewritten as two separate inequalities:

x + 1 ≥ 2 OR x + 1 ≤ -2

Solving these inequalities:

x ≥ 1 OR x ≤ -3

Graphical Representation: Shade the region to the left of -3 (including -3) and the region to the right of 1 (including 1).

     <-------------------•-------------------o------------------->
     -4  -3  -2  -1   0   1   2   3   4
                 x ≤ -3  OR  x ≥ 1

Identifying the Correct Graph

When presented with a solved inequality and several graphs, carefully examine the following:

• Inequality Symbol: Determine whether the inequality includes or excludes the endpoints. Closed circles (•) indicate inclusion (≤ or ≥), while open circles (o) indicate exclusion (< or >).
• Shaded Region: Identify the region shaded on the graph. It should correspond to the solution set of the inequality.
• Endpoints: Verify that the endpoints of the shaded region match the values obtained from solving the inequality.

By carefully following these steps and understanding the different types of inequalities, you can confidently solve inequalities and correctly identify their graphical representations. Remember practice is key to mastering this crucial algebraic skill. Work through numerous examples, varying the types of inequalities and their complexity, to build your understanding and confidence. The more you practice, the easier it will become to identify the correct graph representing the solution to any inequality you encounter.