Problem 43 - Algebra for College Students

Question

Describe in your own words the process of solving a system of inequalities.

Step-by-Step Solution

Answer

Graph the solutions of each inequality and find where they overlap.

Step 1: Understand the Inequalities

First, read and understand each inequality given in the system. Inequalities are similar to equations, but instead of an equal sign, they use symbols like <, >, ≤, or ≥. Make sure you identify the variables and the respective inequality signs correctly.

Step 2: Solve Each Inequality

Solve each inequality individually. To solve an inequality, perform similar operations as you would when solving equations: add, subtract, multiply, or divide both sides of the inequality. Remember, if you multiply or divide both sides by a negative number, you must reverse the inequality sign.

Step 3: Graph Each Solution on a Number Line (if one variable)

For one-variable inequalities, plot the solutions of each inequality on a number line. Use an open circle for < or > and a closed circle for ≤ or ≥. The solution is the shaded region or regions on the number line.

Step 4: Graph Each Solution on a Coordinate Plane (if two variables)

For two-variable inequalities, graph the solutions on a coordinate plane. First, solve for 'y' if necessary to get the inequality into the form y < or y >. Use a dashed line for < or > and a solid line for ≤ or ≥. Shade the appropriate side of the boundary line.

Step 5: Identify the Intersection

Determine the combined solution set by identifying where the shaded regions overlap from all inequalities. This intersection represents the solutions that satisfy all inequalities in the system.

Step 6: Write the Solution

Express the solution set either as an interval for one-variable inequalities or as a region within the coordinate plane for two-variable inequalities. Make sure the solution satisfies all inequalities in the system.

Key Concepts

Graphing Inequalities

Understanding how to graph inequalities is crucial for visually interpreting their solutions. For one-variable inequalities, imagine a number line. Here, you'll plot the solution of each inequality using circles and shading:

• An **open circle** is used for < or >, indicating the value isn't included in the solution set.
• A **closed circle** is utilized for ≤ or ≥, showing that the value is part of the solution.

The shaded region on the number line reveals all possible solutions.
For two-variable inequalities, you need a coordinate plane. Start by rearranging the inequality to the "y <" or "y >" form. The graph's boundary is a line:

• **Dashed line** for < or >, excluding points exactly on the line.
• **Solid line** for ≤ or ≥, including those points.

Then, shade the half-plane that satisfies the inequality. The overlap of all shaded regions from each inequality is the solution to the system.

Solving Inequalities

Solving inequalities is quite similar to solving equations, but there's a special twist concerning the inequality signs. Just like equations, you can:

• Add or subtract the same amount from both sides.
• Multiply or divide both sides by a positive number.

However, a unique rule applies when multiplying or dividing by a negative number: the inequality sign must be flipped. For instance:

• If \( x < 5 \), and you multiply both sides by -1, it becomes \( -x > -5 \).

This operation rule makes inequalities a bit tricky, so always watch out when negative numbers are involved. After solving each inequality separately, you are ready to graph their solutions or find where solutions intersect.

Inequality Symbols

Inequality symbols might seem simple, but they carry significant meaning and influence the interpretation of inequality expressions. The primary inequality symbols are:

• < (less than): Indicates that the number on the left is smaller than the number on the right.
• > (greater than): Shows the number on the left is larger than the number on the right.
• ≤ (less than or equal to): Means the number on the left is either smaller or exactly equal to the right number.
• ≥ (greater than or equal to): States that the number on the left is either bigger or precisely the same as the number on the right.

These symbols dictate how inequalities are graphed, whether open or closed circles are used, and how the boundary lines should be drawn in graphs with two variables. Recognizing and properly applying these symbols is essential for solving and graphing inequalities accurately.