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Chapter 6: Problem 32

Solve the inequality. $$2(x-4) \geq 3$$

Short Answer

Expert verified
The solution is \(x \geq 5.5\).

Step by step solution

01

Expand the left side

The first step is to distribute the '2' to both 'x' and '-4'. This will result in \(2x - 8 \geq 3\)
02

Isolate the variable

Next, add 8 to both sides of the inequality to isolate 2x. This results in \(2x \geq 11\)
03

Solve for x

To find the value of 'x', you need to divide both sides by '2'. This gives \(x \geq \frac{11}{2}\) or \(x \geq 5.5\)

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Key Concepts

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

Distributive Property
When solving inequalities, one of the first techniques you might use is the distributive property. This property helps you distribute a number across an expression within parentheses. For an expression like \(2(x-4)\), the distributive property allows you to multiply '2' with each term inside the parentheses.

This results in:
  • \(2 \cdot x = 2x\)
  • \(2 \cdot (-4) = -8\)
So, the expression becomes \(2x - 8\). By doing this, you simplify equations and inequalities, making it easier to work with each term. The distributive property is a foundational tool in algebra, useful for dealing with both equations and inequalities.
Isolation of Variable
The next step in solving inequalities is isolating the variable. This means arranging the inequality so that the variable you're solving for, like \(x\), is by itself on one side of the inequality. Let's take a closer look.

In our example, after using the distributive property, you have \(2x - 8 \geq 3\). To isolate \(2x\), you need to add '8' to both sides. It is crucial to perform the same operation on each side to maintain the balance of the inequality.
Performing the calculation:
  • Left side: \(2x - 8 + 8 = 2x\)
  • Right side: \(3 + 8 = 11\)
So, your inequality becomes \(2x \geq 11\). Now, \(2x\) is isolated, and getting closer to solving for \(x\). This step is essential in moving towards discovering the value of the variable.
Division in Inequalities
After isolating the variable with its coefficient, the next step involves division to find the variable's value. The inequality \(2x \geq 11\) needs you to get \(x\) by itself. You accomplish this by dividing both sides by the coefficient of \(x\), which is '2'.

Here's how the division works:
  • Left side: \(\frac{2x}{2} = x\)
  • Right side: \(\frac{11}{2} = 5.5\)
Thus, the inequality transforms into \(x \geq 5.5\). It's critical to note that if you were dividing by a negative number, you would need to flip or reverse the inequality sign. However, since we're dividing by a positive number, the inequality sign remains the same. As a result, you've successfully solved the inequality and found that \(x\) is greater than or equal to 5.5.

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Most popular questions from this chapter

Solve the inequality. Then graph and check the solution. $$ |-3+x|<18 $$

Write the equation corresponding to the inequality in slope-intercept form. Tell whether you would use a dashed line or a solid line to graph the inequality. $$ -4 x-2 y<6 $$

Write an inequality that represents the statement. x is greater than or equal to -4 and less than or equal to 4.

Solve the equation. (Lessons 3.1, 3.2) $$x+17=9$$

In 1967 a 60-second television commercial during the first Super Bowl cost \(\$85,000\). In 1998 advertisers paid \(\$2.6\) million for two 30-second spots. Assuming those were the least and greatest costs during that period, write an inequality that describes the cost c of 60 seconds of commercial time from 1967 to 1998.
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