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Chapter 2: Problem 32

Solve each of the inequalities and graph the solution set on a number line. \(4-2 x<12\)

Short Answer

Expert verified
The solution is \(x > -4\).

Step by step solution

01

Isolate the Variable Term

We start by isolating the variable term on one side of the inequality. To do this, we'll subtract 4 from both sides:\[-2x < 12 - 4\]Simplifying the right-hand side, we get:\[-2x < 8\]
02

Solve for the Variable

Next, we need to solve for \(x\) by dividing both sides by \(-2\). Remember, when you divide or multiply by a negative number, the inequality sign flips:\[x > \frac{8}{-2}\]Simplify the right-hand side:\[x > -4\]
03

Graph the Solution on a Number Line

To graph the solution \(x > -4\) on a number line, draw a number line and indicate \(-4\) on it. Since \(x\) is greater than \(-4\), use an open circle at \(-4\) and shade to the right to show all numbers greater than \(-4\). This represents our solution set.

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Key Concepts

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

Algebra
Algebra provides us with the tools to handle equations and inequalities involving variables. When dealing with inequalities like the one in our exercise, which is \(4 - 2x < 12\), we aim to find the set of all possible values for \(x\) that make this statement true. The key steps involve isolating the variable term and performing operations to simplify the expression. This includes:
  • Adding or subtracting terms from both sides
  • Dividing or multiplying both sides by a constant
  • Remembering that the direction of the inequality changes when both sides are multiplied or divided by a negative number
By understanding these operations, algebra enables us to transition from complex problems to simpler expressions that are easier to interpret and solve.
Graphing Inequalities
Graphing inequalities is a visual way of depicting the solution set of an inequality. After solving the inequality algebraically, like \(x > -4\) in our problem, we can represent this solution on a graph.

Here's how to do it:
  • Draw a horizontal line, representing a number line.
  • Mark the specific point, in this case, \(-4\).
  • Use an open circle for \(-4\) since values equal to \(-4\) do not satisfy \(x > -4\).
  • Shade the region to the right of \(-4\), indicating all numbers greater than \(-4\) are part of the solution set.
This shaded area visually represents all possible values of \(x\) that satisfy the inequality, helping students easily understand and interpret the solution.
Number Line
A number line is a simple yet powerful tool for visualizing mathematical concepts, providing a clear way to represent numbers and inequalities physically. When graphing on a number line, pay attention to:
  • Integer placement and proportional spacing.
  • Using open or closed circles to represent strict \((>)\) or inclusive \((\geq)\) inequalities.
  • Direction of shading to indicate solution sets.
In the context of inequalities, a number line serves to illustrate the range of solutions clearly. For example, for \(x > -4\), the open circle at \(-4\) and shading to the right instantly communicate which values \(x\) can take. This visual method not only aids in comprehension but also helps to confirm the accuracy of the algebraic solution.

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Most popular questions from this chapter

For Problems \(35-44\), solve each compound inequality and graph the solution sets. Express the solution sets in interval notation. $$ x-4<-2 \text { or } x-4>2 $$

For Problems \(17-30\), solve each inequality and graph the solution. $$ |x-2|<4 $$

For Problems 45-56, solve each compound inequality using the compact form. Express the solution sets in interval notation. $$ -3<2 x+1<5 $$

For Problems 45-56, solve each compound inequality using the compact form. Express the solution sets in interval notation. $$ -17 \leq 3 x-2 \leq 10 $$

The temperatures for a 24-hour period ranged between \(-4^{\circ} \mathrm{F}\) and \(23^{\circ} \mathrm{F}\), inclusive. What was the range in Celsius degrees? \(\left(\right.\) Use \(F=\frac{9}{5} C+32\). \()\)
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