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Chapter 1: Problem 13

Solve each inequality. Graph the solution. \(-7(3 x-7)+21 x \geq 50\)

Short Answer

Expert verified
The inequality has no solution, and the graph is empty.

Step by step solution

01

Distribute the multiplication

First, simplify the left side of the inequality by distributing the multiplication of -7 through the parenthesis: -7 * (3x - 7) = -21x + 49. The inequality becomes: -21x + 49 + 21x ga 50.
02

Combine like terms

On the left side of the inequality, combine the terms that contain x: -21x + 21x, which cancel each other out. The inequality now simplifies to: 49 ga 50.
03

Analyze the simplified inequality

The inequality now reads 49 ga 50, which is not true since 49 is not greater than or equal to 50. Since no x can satisfy this inequality, there are no solutions.
04

Graph the solution

Since there are no values of x that can satisfy the inequality, the graph would be an empty graph, indicating that no portion of the x-axis satisfies the original inequality.

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Key Concepts

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

Inequality Graphing
Inequality graphing is a visual way to represent the solutions set to an inequality on a number line or coordinate plane. It helps students to better understand which numbers are included in the set of solutions.

When graphing the solution to an inequality such as \( -7(3x-7)+21x \geq 50 \), it's essential to first solve the inequality. If the inequality is true for all values of \(x\), you would shade the entire number line. If it's true for no values of \(x\), as in the example given, the number line stays unshaded, indicating no solution.
Combining Like Terms
Combining like terms is an essential step in simplifying mathematical expressions and inequalities. Like terms are terms that have exactly the same variable parts, such as \(2x\) and \(3x\), and can be added or subtracted from each other.

In the example \( -7(3x-7)+21x \geq 50 \), after distributing the multiplication, we combine \( -21x \) and \( +21x \) because they are like terms. This effectively cancels out the \( x \) terms, which leads us to a simplified inequality without variables.
Mathematical Inequalities
Mathematical inequalities express the relation between two expressions that may not be equal. Symbols such as <, >, 鈮�, and 鈮� denote these relationships. When an inequality has no solution, it means that there are no possible values of the variable that make the inequality true.

For instance, after simplifying the expression \( -7(3x-7)+21x \geq 50 \), we obtain \( 49 \geq 50 \), which is not true for any value of \(x\). Therefore, we say the inequality has no solution.
Algebraic Steps
Solving inequalities often involves several algebraic steps: distributing, combining like terms, isolating the variable, and sometimes even dividing or multiplying through by a negative (which flips the inequality sign).

Throughout the process, maintaining the balance of the inequality is crucial. It's much like solving an equation, each step intentionally transforming the original problem into simpler equivalents, until we reach a conclusion about the set of solutions, as seen in the provided example.

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Most popular questions from this chapter

Suppose you roll a number cube. Find each theoretical probability. \(P\)(a number greater than 5)

Solve each compound inequality. Graph the solutions. \(36 \geq 1-5 z>-21\)

Which number is a solution of \(|x-3|=x-3 ?\) $$ \begin{array}{llll}{\text { A. }-3} & {\text { B. } 0} & {\text { C. } 1} & {\text { D. } 3}\end{array} $$

Solve each absolute value inequality. \(2|x+3| \geq 10\)

Justify each step by identifying the property used. \(\begin{aligned} \frac{1}{2}(y+3) &>\frac{1}{3}(4-y) \\ 3(y+3) &>2(4-y) \\ 3 y+9 &>8-2 y \\ 5 y+9 &>8 \\ 5 y &>-1 \\ y &>-0.2 \end{aligned}\)
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