Problem 48 - Precalculus Student Solutions Manual 5th

Question

Solve each equation or inequality. $$|5 x+2|-2<3$$

Step-by-Step Solution

Answer

The solution to the inequality is \[-\frac{7}{5} < x < \frac{3}{5}.\]

Step 1 - Move constant outside the absolute value

Add 2 to both sides of the inequality: \[|5x + 2| - 2 + 2 < 3 + 2\]Simplifies to:\[|5x + 2| < 5\]

Step 2 - Create compound inequalities

Since the absolute value inequality \(|A| < B\) means that \[- B < A < B,\]we write: \[-5 < 5x + 2 < 5\]

Step 3 - Solve for x: Part 1

Begin with the left portion of the compound inequality:\[-5 < 5x + 2\]Subtract 2 from both sides:\[-7 < 5x\]Then divide by 5:\[x > -\frac{7}{5}\]

Step 4 - Solve for x: Part 2

Next, solve the right portion of the inequality:\[5x + 2 < 5\]Subtract 2 from both sides:\[5x < 3\]Then divide by 5:\[x < \frac{3}{5}\]

Step 5 - Combine the results

Combine the two inequalities:\[-\frac{7}{5} < x < \frac{3}{5}\]

Key Concepts

compound inequalities

Compound inequalities involve combining two inequalities into one statement separated by the word 'and' or 'or.' For example, in solving the inequality \[|5x + 2| - 2 < 3\], we converted it into a compound inequality. Starting from \[|5x + 2| < 5\], which means: \[-5 < 5x + 2 < 5\].
The 'and' in \[-5 < 5x + 2\] and \[5x + 2 < 5\] signifies that both conditions must be true at the same time. This is a typical characteristic of compound inequalities.
By splitting the compound inequality, we create two separate inequalities to solve: \[-5 < 5x + 2\] and \[5x + 2 < 5\]. Each segment is simplified step-by-step to isolate the variable x. After solving, these inequalities are combined again to show the range of possible values for x: \[-\frac{7}{5} < x < \frac{3}{5}\].

solving inequalities

Inequalities are mathematical statements that show the relationship of one quantity being greater or less than another. To solve them, similar rules and steps apply as they do for solving equations, with a few key differences.
In the example \[|5x + 2| - 2 < 3\], our goal is to isolate the variable. We started by moving constants outside the absolute value: \[|5x + 2| < 5\].
Next, we split the absolute value inequality into two parts, creating a compound inequality: \[-5 < 5x + 2 < 5\].
Simplify each part:

• For \[-5 < 5x + 2\], subtract 2 from both sides getting \[-7 < 5x\], and then divide by 5 yielding \[x > -\frac{7}{5}\].
• For \[5x + 2 < 5\], subtract 2 from both sides getting \[5x < 3\], and then divide by 5 yielding \[x < \frac{3}{5}\].

Combining these results gives the final solution: \[-\frac{7}{5} < x < \frac{3}{5}\].
Remember, when multiplying or dividing both sides of an inequality by a negative number, the inequality sign must be reversed.

precalculus problems

Precalculus problems often involve equations and inequalities that require a clear understanding of algebraic principles. One essential aspect is handling absolute values and inequalities effectively.
In the problem \[|5x + 2| - 2 < 3\], solving it requires knowing how to manage absolute values and create compound inequalities. The step-by-step breakdown is crucial:

• First, add 2 to move the constant outside the absolute value, simplifying to \[|5x + 2| < 5\].
• Convert this absolute value into a compound inequality: \[-5 < 5x + 2 < 5\].
• Separate and solve each side of the compound inequality.
• Combine the results to find the solution for x: \[-\frac{7}{5} < x < \frac{3}{5}\].

Understanding these steps ensures a strong foundation not only for solving precalculus problems but also for more complex calculus concepts later.

Practice is key. Try more problems with absolute values and compound inequalities to gain confidence.