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Chapter 2: Problem 30

Solve each of the inequalities and graph the solution set on a number line. \(1+6 x>-17\)

Short Answer

Expert verified
The solution is \(x > -3\).

Step by step solution

01

Isolate the variable term

To begin solving the inequality \(1 + 6x > -17\), we start by isolating the term with variable \(x\) by subtracting 1 from both sides of the inequality. This gives us:\[ 6x > -17 - 1 \] which simplifies to:\[ 6x > -18 \]
02

Solve for x

With the variable term isolated, the next step is to solve for \(x\) by dividing both sides of the inequality by 6:\[ x > \frac{-18}{6} \] which simplifies to:\[ x > -3 \]
03

Graph the solution on a number line

To represent \(x > -3\) on a number line, draw a number line and make an open circle at \(-3\) to indicate that \(-3\) is not included in the solution. Then, shade the number line to the right of \(-3\) to show all the numbers greater than \(-3\) are part of the solution set.

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Key Concepts

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

Isolating the Variable
When you isolate a variable in an inequality, you aim to get the variable by itself on one side of the inequality sign. This is akin to simplifying an equation, but you need to be mindful of inequality properties. Here, in the example of the inequality \(1 + 6x > -17\), we first subtract 1 from both sides to remove the constant term on the left. This methodical subtraction changes the inequality to:
  • \(6x > -17 - 1\), which simplifies to \(6x > -18\)
The essence of isolating the variable is to simplify the equation while maintaining a balanced operation on both sides of the inequality. It is critical to perform the same arithmetic operation on both sides to preserve the inequality's direction and validity.
Graphing Inequalities
Graphing inequalities on a number line provides a visual representation of the solution set. After solving the inequality \(x > -3\), we graph the solution to articulate which values of \(x\) make the inequality true. Start by drawing a horizontal line鈥攖he number line.
  • Place an open circle at \(-3\) since \(x\) is greater than \(-3\) but does not include \(-3\) itself.
  • Shade the line extending to the right of the open circle to indicate all numbers greater than \(-3\) are included.
The open circle denotes that the boundary number itself is not part of the solution, differing from a filled circle that would include the number itself.
Number Line Representation
A number line is an essential tool for visually representing inequalities. It allows you and others to quickly assess which numbers satisfy the inequality. For the expression \(x > -3\), here's how to represent it on the number line:
  • The number line involves evenly spaced tick marks at integer values.
  • An open circle at \(-3\) indicates that \(-3\) is not part of the solution set, which aligns with the expression \(x > -3\).
  • The shading to the right signifies that all values greater than \(-3\) comply with our inequality.
Using a number line helps clarify the difference between inequality types by choosing open or closed circles and directions of shading, presenting a clear depiction of a mathematical relationship.

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Most popular questions from this chapter

Suppose that Derwin shot rounds of \(82,84,78\), and 79 on the first four days of a golf tournament. What must he shoot on the fifth day of the tournament to average 80 or less for the five days?

For Problems \(31-54\), solve each inequality. $$ |x-3|>9 $$

For Problems \(35-44\), solve each compound inequality and graph the solution sets. Express the solution sets in interval notation. $$ 3 x+2>17 \text { and } x \geq 0 $$

For Problems 45-56, solve each compound inequality using the compact form. Express the solution sets in interval notation. $$ 0<2 x+5<12 $$

For Problems 45-56, solve each compound inequality using the compact form. Express the solution sets in interval notation. $$ -1 \leq \frac{x+2}{4} \leq 1 $$
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