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Chapter 5: Problem 7

Solve each inequality and graph the solutions on a number line. a. \(3 x-2 \leq 7\) b. \(4-x>6\) c. \(3+2 x \geq-3\) d. \(10 \leq 2(5-3 x)\)

Short Answer

Expert verified
a: x 鈮� 3, b: x < -2, c: x 鈮� -3, d: x 鈮� 0.

Step by step solution

01

Solve Inequality a

First, we have the inequality \(3x - 2 \leq 7\). To solve for \(x\), add 2 to both sides to get \(3x \leq 9\). Then, divide both sides by 3 to isolate \(x\), resulting in \(x \leq 3\).
02

Graph Solution a

On a number line, plot a closed circle at 3 and shade to the left, indicating all numbers less than or equal to 3 are included in the solution set.
03

Solve Inequality b

The inequality is \(4 - x > 6\). Subtract 4 from both sides to get \(-x > 2\). Then, multiply both sides by -1 (remember to flip the inequality sign) to find \(x < -2\).
04

Graph Solution b

On a number line, plot an open circle at -2 and shade to the left, indicating all numbers less than -2 are in the solution set.
05

Solve Inequality c

For the inequality \(3 + 2x \geq -3\), subtract 3 from both sides to get \(2x \geq -6\). Divide both sides by 2 to get \(x \geq -3\).
06

Graph Solution c

On a number line, plot a closed circle at -3 and shade to the right, indicating all numbers greater than or equal to -3 are part of the solution.
07

Solve Inequality d

The inequality is \(10 \leq 2(5 - 3x)\). First, distribute the 2 on the right side to get \(10 \leq 10 - 6x\). Subtract 10 from both sides to find \(0 \leq -6x\). Finally, divide both sides by -6 and remember to flip the inequality sign, resulting in \(x \leq 0\).
08

Graph Solution d

On a number line, plot a closed circle at 0 and shade to the left, indicating all numbers less than or equal to 0 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.

Graphing Inequalities
Graphing inequalities helps visualize solution sets on a number line, making it easier to understand which values of variable make the inequality true. To graph an inequality:
  • First, solve the inequality to determine the boundary point. This is where you'll place a circle on the number line.

  • Use an open circle if the inequality is strict (e.g., < or >) because it does not include the boundary point.

  • Use a closed circle if the inequality is inclusive (e.g., 鈮� or 鈮�) because it includes the boundary point.

  • Shade the region of the number line that represents the solution, in the direction that satisfies the inequality.
Graphing helps students not only find solutions but also see the range of possible solutions quickly. It turns abstract inequalities into visible and intuitive lines.
Number Line
A number line is a visual tool where numbers are laid out in a straight horizontal line, but can also be vertical. Each point on the line represents a number. This helps understand the order and spacing between numbers:
  • Positive numbers are to the right of zero, and negative numbers to the left.

  • A number's position on the line shows its value relative to other numbers.

  • Number lines are useful in inequalities to show which sets of values satisfy the inequality conditions.

  • Plotting inequalities involves marking a specific point with a circle and shading a region to express the range of solutions.
Number lines make inequalities approachable by connecting algebraic expressions to a spatial concept. They help deepen understanding of numerical relationships.
Algebraic Expressions
Algebraic expressions are mathematical phrases that can include numbers, variables, and operation symbols. They form the backbone of equations and inequalities:
  • Variables, like \( x \), represent unknown values that we solve for in an inequality.

  • Operations such as addition, subtraction, multiplication, and division help manipulate expressions to isolate the variable.

  • Understanding and simplifying expressions is key to solving inequalities. This can be done by performing the same operation on both sides of the inequality to maintain its balance.

  • Algebraic expressions can represent real-world problems and situations, turning words into solvable mathematical equations.
Grasping how to work with algebraic expressions is fundamental in math. They are the tools that allow students to translate and solve complex problems.

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Most popular questions from this chapter

APPLICATION The manager of a movie theater wants to know the number of adults and children who go to the movies. The theater charges \(\$ 8\) for each adult ticket and \(\$ 4\) for each child ticket. At a showing where 200 tickets were sold, the theater collected \$1304. a. Let the variable \(A\) represent the number of adult tickets and \(C\) represent the number of child tickets. Write an equation for the total number of tickets sold. (ia) b. Write an equation showing the total cost of the tickets. (a) c. Use your equations from \(9 \mathrm{a}\) and \(\mathrm{b}\) to write a system whose solution represents the number of adult and child tickets sold. Solve this system by symbolic manipulation.

Graph this system of inequalities on the same set of axes. Describe the shape of the region. $$ \left\\{\begin{array}{l} y \leq 4+\frac{2}{3}(x-1) \\ y \leq 6-\frac{2}{3}(x-4) \\ y \geq-17+3 x \\ y \geq 1 \\ y \geq 7-3 x \end{array}\right. $$

List the order in which you would perform these operations to get the correct answer. a. \(72-12 \cdot 3.2=33.6\) b. \(2+1.5\left(3-5^{2}\right)=-31\) c. \(21 \div 7-6 \div 2=0\)

Sketch a graph showing the solution to each system. a. \(y \leq 2\) b. \(x+y \leq 4\) \(x<2\) \(x-y \leq 4\)

APPLICATION On Monday a group of students started on a three-day bicycle tour covering a total of \(286 \mathrm{~km}\). On Tuesday they cycled \(7 \mathrm{~km}\) less than on Monday. On Wednesday they traveled \(24 \mathrm{~km}\) less than on Tuesday. a. Write a system of three linear equations representing this trip. Use \(m, t\), and \(w\) to represent the distances in kilometers they cycled on Monday, Tuesday, and Wednesday, respectively. Write each equation in the form \(a m+b t+c w=d\). (a) b. Write a \(3 \times 4\) matrix to model this system of equations. Describe what the rows and columns of your matrix represent. (a) c. List and describe a sequence of matrix row operations that will produce a matrix of the form $$ \left[\begin{array}{llll} 1 & 0 & 0 & ? \\ 0 & 1 & 0 & ? \\ 0 & 0 & 1 & ? \end{array}\right] $$ d. What is the solution to this problem?
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