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Chapter 3: Problem 33

Solve and graph the inequality. $$4-2 x<3$$

Short Answer

Expert verified
The solution to the inequality is \(x > 0.5\). This represents all numbers greater than 0.5.

Step by step solution

01

Subtract 4 from Both Sides

Subtract 4 from both sides of the inequality. That gives: \(-2x < 3 - 4\), which simplifies to: \(-2x < -1\).
02

Divide by -2

Divide both sides by -2 to isolate \(x\). Note that when we divide or multiply an inequality by a negative number, the direction of the inequality sign changes. Therefore dividing \(-2x < -1\) by -2 leads to \(x > 0.5\).
03

Graph the Inequality

On a number line, make a hole at 0.5 (because \(x\) is not equal to 0.5) and shade everything to the right of 0.5. This is the graphical solution to the inequality.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Graphing Inequalities
When graphing inequalities, you represent the solutions visually on a number line. This allows you to see all possible solutions at a glance. Inequalities such as \(x > 0.5\) involve showing where all values greater than 0.5 exist on a number line.
To graph, follow these steps:
  • Identify the critical number, here it's 0.5.
  • Draw a number line and mark this point.- Because this is a "greater than" inequality (\(x > 0.5\)), use an open circle at 0.5. This indicates that 0.5 is not included in the solution.
  • Shade the region to the right of 0.5. This shows all numbers larger than 0.5 satisfy the inequality.
The key to successful graphing lies in accurately determining the boundary point, selecting the correct type of circle, and knowing which direction to shade.
Algebraic Manipulation
Algebraic manipulation refers to rearranging and simplifying expressions to solve equations or inequalities. In our problem, \(4 - 2x < 3\), the goal is to have the variable \(x\) isolated on one side. Begin by subtracting 4 from both sides, which transforms the inequality to \(-2x < -1\).
Next, manage the variable's coefficient. The division by a negative number invokes a crucial rule: flipping the inequality sign. So, dividing both sides of \(-2x < -1\) by -2 results in \(x > 0.5\).
  • Always perform the same operation on both sides to maintain balance.
  • Remember that dividing or multiplying by a negative flips the inequality.
  • Check your solution by substituting values back into the original inequality.
Mastering algebraic manipulation requires understanding properties of equality and inequalities, and constant practice.
Number Line Representation
A number line provides a clear visual aid for understanding inequalities. It helps you easily see solution sets and relationships between numbers. To effectively use a number line:
  • Begin by drawing a straight horizontal line.
  • Mark equal intervals to represent numbers.
  • Identify the points of interest, such as 0.5 in our inequality \(x > 0.5\).
  • Place an open circle at 0.5 to indicate it is not part of the solution.
Shading, or using arrows, indicates the range of possible solutions. In this exercise, shading to the right of the open circle signifies all numbers greater than 0.5. Number lines are incredibly helpful visual tools that support understanding mathematical relationships.

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