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Chapter 4: Problem 27

Graph the solution of each inequality on a number line. $$-\frac{3 p}{4}<6$$

Short Answer

Expert verified
p > -8

Step by step solution

01

- Isolate the variable term

The goal is to solve for the variable p. To do this, start by isolating the term with p. Since the term is being multiplied by \(-\frac{3}{4}\), the first step is to eliminate the fraction by multiplying both sides of the inequality by \(-\frac{4}{3}\). Note that multiplying or dividing both sides of an inequality by a negative number reverses the inequality sign.
02

- Perform multiplication

Multiply both sides of the inequality by \(-\frac{4}{3}\): \[-\frac{3p}{4} \cdot -\frac{4}{3} > 6 \cdot -\frac{4}{3}\]. Simplifying gives \p > -8\.
03

- Graph the solution on a number line

To graph the solution on a number line, draw an open circle at \(-8\) to indicate that \(-8\) is not included. Then, shade the number line to the right of \(-8\) to indicate that the solution includes all values greater than \(-8\).

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Key Concepts

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

Isolating Variables in Inequalities
Isolating the variable in an inequality is crucial to finding its solution. It means getting the variable by itself on one side of the inequality symbol.

In the given example, \(\frac{-3p}{4} < 6\), the term we need to isolate is **p**. Since the term is being multiplied by \(-\frac{3}{4}\), the first step is to eliminate this fraction by performing the inverse operation, which is multiplying both sides by its reciprocal, \( -\frac{4}{3}\).

When isolating variables, remember to maintain the balance of the inequality. Perform the same operation on both sides to keep it true. This operation sets the stage for isolating \(p\) and finding its range of values.
Number Line Representation
Graphing inequalities on a number line helps visualize their solutions easily. After determining the solution range, we graphically represent it. In the provided example, the solution was found to be \(p > -8\).

To graph this:
  • Draw a number line with appropriate scaling.
  • Place an open circle at \(-8\) to indicate that \(-8\) is not included in the solution.
  • Shade the number line to the right of \(-8\) to show that all values greater than \(-8\) are part of the solution set.

Visual representations like these make it clear and understandable where the solution range lies.
Multiplying Inequalities by Negative Numbers
When solving inequalities, multiplication or division by negative numbers requires special attention because it reverses the inequality sign. This fundamental rule ensures the inequality remains true after the operation.

In our example, multiplying both sides of the inequality \(\frac{-3p}{4} < 6\) by \(-\frac{4}{3}\) involves a negative number. To correctly apply the rule:
  • Multiply both sides by \(-\frac{4}{3}\): \left(\frac{-3p}{4}\right) \times \left(-\frac{4}{3}\right) > 6 \times \left(-\frac{4}{3}\right). \
  • The inequality sign is reversed from 'less than' to 'greater than'. \ \text{This results in } p > -8\.

This step reinforces the importance of changing the inequality's direction whenever you multiply or divide both sides by a negative number, ensuring the solution is accurate and correctly represented.

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Most popular questions from this chapter

Solve each inequality. $$5(e-2)>10$$

Solve the systems of equations in Exercises \(12-15\) by elimination, and check your solutions. Give the following information: \(\cdot\) which equation or equations you rewrote \(\cdot\) how you rewrote each equation \(\cdot\) whether you added or subtracted equations \(\cdot\) the solution $$ \begin{array}{ll}{3 m+n=7} & {[\mathrm{A}]} \\ {m+2 n=9} & {[\mathrm{B}]}\end{array} $$

The equation \(x^{3}+5 x^{2}+4=5\) has a solution between \(x=0\) and \(x=1 .\) Find this solution to the nearest tenth.

Graph the solution of each inequality on a number line. $$12-5 q \geq 32$$

In Exercises \(11-14,\) write and solve an equation to find the number of coins each friend has. Ken has three more coins than twice the number Javier has. Khalid has five fewer coins than Javier. They have 50 coins altogether.
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