Problem 21 - Algebra 2

Question

Solve each inequality. Then graph the solution set on a number line. \(90 \geq 5(2 r+6)\)

Step-by-Step Solution

Answer

The solution is \(r \leq 6\) and is graphed with a closed circle at 6 and shading to the left.

Step 1: Simplify the Inequality

First, distribute the 5 on the right side of the inequality to both terms inside the parentheses. We have:\[90 \geq 5(2r + 6) \90 \geq 10r + 30\]

Step 2: Isolate the Variable Term

Subtract 30 from both sides to isolate the term with the variable \(r\):\[90 - 30 \geq 10r + 30 - 30 \60 \geq 10r\]

Step 3: Solve for the Variable

Now, divide each side of the inequality by 10 to solve for \(r\):\[\frac{60}{10} \geq \frac{10r}{10} \6 \geq r\]

Step 4: Write the Solution in Inequality Form

The inequality \(6 \geq r\) means that \(r\) can be any real number less than or equal to 6. This can also be written as \(r \leq 6\).

Step 5: Graph the Solution on a Number Line

To graph \(r \leq 6\) on a number line, draw a line and shade to the left from 6, then put a closed circle on 6 to indicate that 6 is included in the solution.

Key Concepts

Graphing Solutions

Graphing solutions of inequalities is an important way to understand the range of possible values that satisfy an inequality. When we solve an inequality, such as \(r \leq 6\), we aren't looking for just one solution but a whole set of solutions. This set of solutions is often presented visually on a number line to make it easy to see all values that satisfy the inequality.

• Identify the critical point. For \(r \leq 6\), the critical point is 6.
• Decide whether the critical point is included. Here, it is included, so we use a closed circle.
• Determine the direction for shading. Since it's "less than or equal to," we shade to the left of 6. This shows all numbers less than or equal to 6 as solutions.

So, when graphing solutions, you’re essentially showing a range of numbers that make the inequality true on a number line, reinforcing the concept that inequalities represent many possible solutions.

Distributive Property

The distributive property is a fundamental concept in algebra and is crucial when solving equations or inequalities. It allows you to multiply a single term over terms inside a parenthesis, which simplifies expressions and equations.

• The basic form of the distributive property is \(a(b + c) = ab + ac\).
• For the inequality \(90 \geq 5(2r + 6)\), apply the distributive property to distribute 5 into the terms inside the parenthesis: \(5 \times 2r + 5 \times 6 = 10r + 30\).
• This step helps to turn a complex expression into a simpler, more manageable form.

Using the distributive property correctly is the first step in transforming an inequality into a form where further algebraic manipulation is easy, eventually leading to the isolation of the variable.

Isolation of Variables

Isolating the variable is a critical part of solving equations and inequalities. It involves manipulating the equation or inequality so that the variable is on one side by itself. This process allows you to understand what values the variable can take.

• After using the distributive property in \(90 \geq 10r + 30\), the next step is to subtract 30 from both sides: \(90 - 30 \geq 10r\), which simplifies to \(60 \geq 10r\).
• Next, divide both sides by 10 to isolate \(r\): \(\frac{60}{10} \geq r\) becomes \(6 \geq r\).
• This process strips the inequality down to a simple statement about \(r\), making it clear which numbers \(r\) can be.

Isolating variables helps to convert complex inequalities into their simplest forms, making it easier to interpret and solve them.

Number Line Representation

A number line is an excellent visual tool for representing inequalities. It helps you see at a glance which numbers satisfy the inequality.

• Draw a straight line and mark numbers on it, focusing on the critical point, here it's 6.
• Use a closed circle to represent \(6\) because the inequality is \(r \leq 6\), meaning 6 is included in the solution set.
• Shade the line to the left of 6 to show that all numbers less than or equal to 6 are solutions.

This representation makes it very easy to "read" the solution set of an inequality, especially as problems become more complex. Seeing solutions graphically can enhance understanding by providing a clear, visual context.