Problem 49 - Algebra for College Students

Question

Solve each inequality. Graph the solution set, and write it using interval notation. $$ |5 x+2| \leq 10 $$

Step-by-Step Solution

Answer

\[-14 \leq x \leq 10\). In interval notation: \[-14, 10\).

Step 1 - Subtract 5 from all parts

To isolate the variable x, subtract 5 from each part of the inequality: \[ -9 - 5 \leq x + 5 - 5 \leq 15 - 5 \] This simplifies to: \[ -14 \leq x \leq 10 \]

Step 2 - Write the solution in interval notation

The inequality \(-14 \leq x \leq 10\) states that x can be any value from -14 to 10, inclusive. Therefore, the interval notation for the solution is: \[-14, 10\]

Step 3 - Graph the solution set

On a number line, represent the solution set by shading the region from -14 to 10 and placing closed circles (filled dots) at -14 and 10 to indicate that these endpoints are included in the solution set.

Key Concepts

Interval Notation

Interval notation is a shorthand method used to describe subsets of the real number line. It simplifies expressing ranges of values and makes understanding the solutions of inequalities much easier. For example, in our exercise, we arrived at the inequality \[ -14 \le x \le 10 \].
To convert this into interval notation, first identify the smallest (leftmost) and largest (rightmost) values in the solution set. Here, they are -14 and 10. Next, since both endpoints -14 and 10 are included in the solution set, we use closed brackets \[ \] around the numbers. Therefore, the interval notation is \[-14, 10\].
Remember to:

• Use \[ ... \] for inclusive limits (where the inequality symbol is \le or \ge).
• Use \( ... \) for exclusive limits (where the inequality symbol is < or >).

Inequality Graphing

Graphing inequalities is an essential step to visualize the range of solutions. It makes abstract mathematical concepts more tangible. To graph the solution set of \(-14 \le x \le 10 \), follow these steps:

• Draw a number line, a horizontal line with evenly spaced numbers.
• Locate the points -14 and 10 on the number line. Place closed dots (filled circles) at these points because the inequality includes these values (indicated by the \le symbol).
• Shade the region between -14 and 10 on the number line. This shows that any number in this range is a solution to the inequality.

Graphing helps verify the solution and provides a clear and visual representation of the range of possible values for x. This is especially helpful in checking your work or explaining the problem to others.

Algebraic Manipulation

Algebraic manipulation is the process of using mathematical operations to transform equations or inequalities, making them easier to solve. Let's break down how we used algebraic manipulation in the given problem:
Step 1: Given \(-9 \le x + 5 \le 15\), the goal is to isolate x. We needed to get rid of the +5 attached to the x. To do this, subtract 5 from all parts of the inequality:
\[-9 - 5 \le x + 5 - 5 \le 15 - 5\]
This simplifies to:
\[-14 \le x \le 10\]
By isolating x, we've turned a complex inequality into a simpler form that’s easier to interpret and solve. The key steps in algebraic manipulation include:

• Applying the same operation to every part of the inequality or equation.
• Maintaining the inequality’s balance by doing the same exact operation on all sides.
• Simplifying the results step-by-step to reach the solution.

Mastering these techniques allows for solving more complex problems efficiently and is foundational in algebra.