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Chapter 1: Problem 20

Solve the linear inequality. Express the solution using interval notation and graph the solution set. $$2(7 x-3) \leq 12 x+16$$

Short Answer

Expert verified
The solution is \(( -\infty, 11 ]\). The solution set is shaded to the left of and includes 11.

Step by step solution

01

Distribute the 2

First, distribute the 2 on the left side of the inequality to both terms inside the parentheses. This changes the inequality from \(2(7x-3) \leq 12x + 16\) to \(14x - 6 \leq 12x + 16\).
02

Move All x Terms to One Side

Next, subtract \(12x\) from both sides to bring all terms involving \(x\) to one side. \(14x - 12x - 6 \leq 16\) simplifies to \(2x - 6 \leq 16\).
03

Isolate the x Term

Add 6 to both sides to isolate the term with \(x\) on one side of the inequality. This results in \(2x \leq 22\).
04

Solve for x

Divide both sides by 2 to solve for \(x\). This results in \(x \leq 11\).
05

Express in Interval Notation

The solution \(x \leq 11\) can be written in interval notation as \(( -\infty, 11 ]\).
06

Graph the Solution Set

To graph this solution, draw a number line. Shade the entire line to the left of 11 and place a closed dot at 11 to indicate that 11 itself is 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.

Understanding Interval Notation
Interval notation is a simple and concise way to represent a range of values, particularly useful when dealing with inequalities. It uses parentheses and brackets to indicate which endpoints are included, allowing us to quickly identify all the numbers that satisfy a given inequality.

In the solution of the inequality \(x \leq 11\), the interval notation is written as \((-\infty, 11 ]\). This tells us:
  • The interval starts from \(-\infty\), meaning there is no lower bound for the values of \(x\).
  • It goes up to 11.
  • A square bracket \([\, ]\) at 11 indicates that 11 is included in the set of solutions.
  • A parenthesis \(( \) at \(-\infty\) denotes that it is not an actual number and cannot be included.
Whenever you see interval notation, you can quickly visualize which numbers are included simply by observing the brackets and parentheses used.
Solving Inequality Solutions
When solving linear inequalities, like \(2(7x-3) \leq 12x+16\), the goal is to find the set of values for \(x\) that make the inequality true. Here is a step-by-step approach to solve such inequalities:
  • Distribute: This involves multiplying the number outside the parenthesis with every term inside to eliminate the parentheses. For instance, \(2(7x-3)\) becomes \(14x - 6\).
  • Combine Like Terms: After removing parentheses, rearrange the terms to get all \(x\)-terms on one side. This might involve adding or subtracting terms from both sides of the inequality.
  • Isolate the Variable: Simplify the inequality to isolate \(x\) on one side. Often, this involves adding, subtracting, multiplying, or dividing both sides by a number. Remember, if you multiply or divide by a negative number, you must flip the inequality sign.
By following these steps, you'll determine the inequality solution, which in this case, is \(x \leq 11\). This tells us that \(x\) can be any number less than or equal to 11, including 11 itself.
Graphing Inequalities Made Easy
Graphing inequalities on a number line is a great way to visually represent the solution set. This process was used to solve \(x \leq 11\) from the given inequality:
  • Draw a Number Line: Begin by drawing a horizontal line and marking key numbers, focusing on the number included in the inequality (in this case, 11).
  • Closed or Open Dot: Use a closed dot at 11 to indicate that 11 is part of the solution set, meaning \(x\) can equal 11.
  • Shading: Shade the line to the left of 11. This shows that all numbers less than 11 and including 11 satisfy the inequality.
Remember, a closed dot denotes the number is included in the solution (as with \(x \leq 11\)), while an open dot would indicate it is not included (as would be the case with \(x < 11\)). Graphing this way helps understand the entirety of the solution set at a glance.

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Most popular questions from this chapter

Use a Special Factoring Formula to factor the expression. $$(x+3)^{2}-4$$

Use the Difference of Squares Formula to factor \(17^{2}-16^{2}\). Notice that it is easy to calculate the factored form in your head, but not so easy to calculate the original form in this way. Evaluate each expression in your head: (a) \(528^{2}-527^{2}\) (b) \(122^{2}-120^{2}\) (c) \(1020^{2}-1010^{2}\) Now use the Special Product Formula $$(A+B)(A-B)=A^{2}-B^{2}$$ to evaluate these products in your head: (d) \(79 \cdot 51\) (e) \(998 \cdot 1002\)

Factor the expression completely. (This type of expression arises in calculus when using the "product rule.") $$\left(x^{2}+3\right)^{-1 / 3}-\frac{3}{3} x^{2}\left(x^{2}+3\right)^{-4 / 3}$$

Show that the equation represents a circle, and find the center and radius of the circle. $$x^{2}+y^{2}+6 y+2=0$$

Factor the expression completely. $$4 x^{2}-25$$
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