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Chapter 9: Problem 12

Solve each linear inequality and graph the solution set on a number line. $$-4(x+2)>3 x+20$$

Short Answer

Expert verified
The solution to the inequality is \(x < -4\). This indicates all the numbers to the left of -4 on the number line are part of the solution set.

Step by step solution

01

Distribute

First, distribute -4 on the left side of the inequality to get rid of the brackets. This gives the new inequality as -4x - 8 > 3x + 20.
02

Simplify the inequality

Now, move the x terms to one side and the integer terms to the opposite side. To do this, add 4x to both sides (to move all x terms to the right) and subtract 20 from both sides (to move the constant term to the left). This gives the new inequality as -28 > 7x.
03

Solve for x

Now, solve for x by dividing through by 7. This gives the result as x < -4.
04

Graph on a number line

Graph the solution set on a number line. Because the inequality sign is '<', use an open circle at -4 and draw an arrow to the left to indicate that all numbers less than -4 are included in the solution set.

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Key Concepts

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

Distributive Property
When you encounter an expression such as \(-4(x + 2)\), this is where the distributive property comes into play. The distributive property is a fundamental concept in algebra that allows us to simplify expressions.
It works by multiplying the number outside the parentheses by each term inside the parentheses. This is like handing out a portion of the outside value to each term inside.
  • Example: In the expression \(-4(x + 2)\),you multiply \(-4\) by \(x\) and also by \(2\).
  • Therefore, \(-4 imes x = -4x\) and \(-4 imes 2 = -8.\)
As a result, the expression becomes \(-4x - 8.\)
This is an essential step in solving inequalities as it simplifies the problem, allowing you to manage and combine like terms later.
Solving Inequalities
Solving inequalities is similar to solving equations, but with a few twists. This process involves getting the variable on one side of the inequality and isolating it to find the solution.
In our example, after using the distributive property, we obtained\[-4x - 8 > 3x + 20.\]To isolate the variable \(x\), follow these steps:
  • Add \(4x\) to both sides of the inequality to combine the \(x\) terms:
    \[-8 > 3x + 4x + 20\]
  • Subtract \(20\) from both sides to get all the constant numbers to the other side:
    \[-8 - 20 > 7x\]
  • Simplify this to:
    \[-28 > 7x\]
Now, divide every term by \(7\)to get \(x\) alone:\[x < -4.\]
Note that when dividing or multiplying both sides of an inequality by a negative number, the inequality sign flips direction. However, in this inequality, the need to flip the sign did not arise.
Graphing on a Number Line
Once you solve an inequality, it's helpful to represent the solution visually on a number line. It allows you to easily see not only the solution but the range of possible solutions as well.
For the inequality \(x < -4\), you represent this on a number line as follows:
  • Locate the number \(-4\) on the number line.
  • Use an open circle at \(-4\) to indicate that \(-4\) is not included in the solution set.
  • Draw an arrow extending to the left of \(-4\) to show that all numbers less than \(-4\) satisfy the inequality.
This graphical representation gives a quick visual overview, making it clear that any number smaller than \(-4\) is a valid solution to the inequality.

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Most popular questions from this chapter

a. Solve: \(x-3<5\) b. Solve: \(2 x+4<14\) c. Give an example of a number that satisfies the inequality in part (a) and the inequality in part (b). d. Give an example of a number that satisfies the inequality in part (a), but not the inequality in part (b).

Describe what is meant by the union of two sets. Give an example.

Will enable you to review graphing linear functions. In addition, they will help you prepare for the material covered in the next section. In each exercise, graph the linear function. $$f(x)=-\frac{2}{3} x \text { or } y=-\frac{2}{3} x(\text { Section } 3.4, \text { Example } 4)$$

Exercises \(99-101\) will help you prepare for the material covered in the first section of the next chapter. If \(f(x)=\sqrt{3 x+12},\) find \(f(8)\)

Write a system of inequalities whose solution set includes every point in the rectangular coordinate system.
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