Problem 25 - Beginning and Intermediate Algebra

Question

Solve each inequality. $$ (4-3 x)^{2} \geq-2 $$

Step-by-Step Solution

Answer

The inequality is true for all x.

Step 1: Understand the Inequality

The given inequality is \((4-3x)^{2} \geq -2\). Recall that any square of a real number is always non-negative. So, \((4-3x)^{2} \geq 0\).

Step 2: Compare to the Given Inequality

Since \((4-3x)^{2}\) is always non-negative, and non-negative numbers are always greater than or equal to \(-2\), the inequality \((4-3x)^{2} \geq -2\) is always true for all x.

Key Concepts

Real Numbers

In mathematics, real numbers include both rational and irrational numbers.
Rational numbers can be expressed as fractions, such as \(\frac{1}{2}\) or \(5\). Irrational numbers cannot be written as simple fractions; examples include \(\frac{\text{pi}}\) and \(\text{sqrt}(2)\).

Real numbers also have the following properties:
- They can be plotted on a number line.
- Every point on the number line corresponds to a real number.
- They are used to measure continuous quantities.

So when working with inequalities involving real numbers, every possible value within a given range needs to be considered.

Non-Negative Numbers

Non-negative numbers are either positive or zero. This can be written as \(x \geq 0\).
Non-negative numbers do not include any negative values. Here are some key points:

• The square of any real number (i.e., \((a^2)\)) is always non-negative.
  
• Applications include measuring distances, since distances cannot be negative.
  
• The inequality \((4-3x)^2 \geq -2\) uses the non-negativity of squares.
  This indicates that our given inequality is always true for all \(x\) because a squared term can't be less than zero, much less a negative number like \(-2\).

Inequality Solution

Solving inequalities involves finding the values of the variable that make the inequality true. For the given problem \((4-3x)^2 \geq -2\), we:
1. Recognize that the square of any real number is non-negative.
2. Compare the squared term to the inequality given: which is always true.

Therefore, for this specific inequality:

• Every real number \(x\) satisfies the inequality \((4-3x)^2 \geq -2\).

In general, to solve a given inequality:

• Isolate the variable on one side.
  
• Use inverse operations to simplify.
• Consider the properties of real numbers, squared terms, and zero.
  Always check your solution set to ensure all values fit the original inequality.