Problem 44 - Algebra for College Students

Question

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

Step-by-Step Solution

Answer

The solution set is (-∞, ∞) and includes all real numbers.

Step 1 - Distribute the terms

Distribute the 7 on the left side and the 2 on the right side of the inequality: 7(4 - x) + 5x < 2(16 - x) 28 - 7x + 5x < 32 - 2x

Step 2 - Combine like terms

Combine the like terms on both sides of the inequality: 28 - 2x < 32 - 2x

Step 3 - Add 2x to both sides

To eliminate the -2x on both sides, add 2x to both sides of the inequality: 28 < 32

Step 4 - Observe the result

The inequality simplifies as 28 < 32, which is always true. Hence, the solution set includes all real numbers.

Step 5 - Write the solution set in interval notation

Since the inequality is always true, the solution set is all real numbers. In interval notation, this is written as (-∞, ∞)

Step 6 - Graph the solution set

The solution set can be graphed on a number line as a line covering all points, indicating that every number is part of the solution set.

Key Concepts

inequality solving steps

Solving an inequality involves a few systematic steps that help to isolate the variable and determine the range of values that satisfy the inequality. Here’s a simplified guide using the example: \(7(4-x)+5x<2(16-x)\).
First, distribute any terms outside parentheses: \(7(4 - x) + 5x < 2(16 - x)\) becomes \(28 - 7x + 5x < 32 - 2x\).
Next, combine like terms on both sides: \(28 - 2x < 32 - 2x\).
Then, you need to isolate the variable. In this case, adding \(2x \) to both sides eliminates it, yielding \(28<32\).
Lastly, analyze the resulting statement. Here, \(28 < 32\), which is always true, showing all real numbers satisfy the inequality.

interval notation

Interval notation is a way of writing subsets of the real number line, often used to express the solution set of an inequality. Using interval notation, we can succinctly express an unbounded range.
In our inequality example, we found that every real number satisfies \(28<32\). This solution set is all real numbers, which we write in interval notation as \((-\infty,\infty)\).
Here are a few tips to remember:

• Use parentheses for inequalities that do not include the endpoints: \((a, b)\) indicates \(a < x < b\).
  
• Use square brackets to include endpoints: \([a, b]\) indicates \(a \leq x \leq b\).
  
• Combine symbols for mixed cases: \((a, b]\) indicates \(a < x \leq b\).
  

For \(\(-\infty, \infty\)\), always use parentheses, as infinity is not a fixed number.

graphing inequalities

Graphing inequalities visualizes the solution set on a number line, showing the range of numbers that satisfy the inequality. Graphing the inequality in our example is straightforward since all real numbers are solutions.
Here’s how to graph solution sets:

• Draw a standard number line.
  
• Identify the solution region. For \((-\infty,\infty)\), the line covers all points.
  
• Shade the appropriate area on the number line. Given all numbers are solutions, shade the entire line.
  

This visual representation helps to quickly see that there are no bounds on the inequality’s solutions.

distributive property

The distributive property helps to eliminate parentheses in mathematical expressions. It states that \(a(b+c)=ab+ac\).
In our problem, we applied it as follows:

• Distribute the 7 across \(4 - x\):\(7(4 - x)=28 - 7x\).
  
• Similarly, distribute the 2 across \(16 - x\): \(2(16 - x)=32 - 2x\).
  

This step is crucial for combining like terms later.

combining like terms

Combining like terms simplifies the expression by merging terms with the same variable. This step makes solving the inequality more manageable.
In the example:

• Combine \(-7x\) and \(5x\) on the left: \(28 - 7x + 5x < 32 - 2x\) becomes \(28 - 2x < 32 - 2x\).
  
• This simplification helps to easily isolate the variable in the next steps.
  

Remember, combining like terms involves adding or subtracting coefficients of the same variable to simplify the expression.