Problem 76 - Algebra for College Students

Question

Solve each equation or inequality. Graph the solution set. $$ 5|x-4|=5 $$

Step-by-Step Solution

Answer

The solutions are x = 3 and x = 5.

Step 1 - Isolate the Absolute Value

Divide both sides of the equation by 5 to isolate the absolute value. This gives:\[ |x-4| = 1 \]

Step 2 - Set Up Two Equations

The absolute value equation |x-4| = 1 can be split into two separate equations:\[ x - 4 = 1 \] and \[ x - 4 = -1 \]

Step 3 - Solve Each Equation

Solve each of the equations from Step 2:For \( x - 4 = 1 \):\[ x = 1 + 4 = 5 \]For \( x - 4 = -1 \):\[ x = -1 + 4 = 3 \]

Step 4 - Graph the Solution

Plot the solutions on a number line. Place a dot at x = 3 and another dot at x = 5, since these are the solutions to the equation 5|x-4| = 5.

Key Concepts

Solving Equations

To solve an absolute value equation, follow these steps. Start by isolating the absolute value expression on one side of the equation. For example, with the equation \( 5|x-4|=5 \), divide both sides by 5 to get \( |x-4| = 1 \).
Now that the absolute value is isolated, set up two separate equations: one where the expression inside the absolute value is equal to the positive value on the other side, and another where it is equal to the negative value.
Using our example, the equations are \( x - 4 = 1 \) and \( x - 4 = -1 \).
Solve each equation separately:

• For \( x - 4 = 1 \): x = 1 + 4 = 5
• For \( x - 4 = -1 \): x = -1 + 4 = 3

Now, you have the solutions: \( x = 3 \) and \( x = 5 \). These are the points where the original equation is true.

Graphing Solutions

Once you have found the solutions for the equation, the next step is to graph them. This helps visualize the solution set, making it easier to understand.
For the example equation \( 5|x-4|=5 \), we found that the solutions are \( x = 3 \) and \( x = 5 \).
To graph these solutions, draw a number line and place a dot at each solution point. In this case, place one dot at \( x = 3 \) and another dot at \( x = 5 \).
This visual representation shows clearly where the solutions to the equation lie on the number line. It can also help identify if there are any patterns, or if the solutions fall within certain intervals, which can be useful for more complicated equations.

Inequalities

An inequality is similar to an equation, but instead of showing equality, it shows that one side is greater or less than the other. When dealing with absolute value inequalities, the process is slightly different from that of absolute value equations.
First, you isolate the absolute value on one side of the inequality. For example, in an inequality \( |x-4| < 1 \), the absolute value is already isolated.
Next, instead of setting up two equalities, you set up two inequalities. For \( |x-4| < 1 \), set up:

• \( x - 4 < 1 \)
• \( x - 4 > -1 \)

Solve each inequality separately:

• For \( x - 4 < 1 \): \( x < 5 \)
• For \( x - 4 > -1 \): \( x > 3 \)

Combine the results to get a compound inequality. Here, it would be \( 3 < x < 5 \).
Graph this solution on a number line, using open circles at \( x = 3 \) and \( x = 5 \), and shade the interval in between to show all the values that satisfy the inequality.
Understanding the graph can help you visualize and confirm the range where the inequality holds true.