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Chapter 5: Problem 29

Graph the solution set. \(-\frac{1}{2} y+4 \leq 5\)

Short Answer

Expert verified
Graph \(y \geq -2\), shading right from \(-2\) with a closed circle at \(-2\).

Step by step solution

01

- Isolate the Variable

Start by isolating the variable term. Add 4 to both sides of the inequality: \(-\frac{1}{2} y + 4 \leq 5 \longrightarrow -\frac{1}{2} y \leq 1\).
02

- Solve for the Variable

Multiply both sides by \(-2\) to solve for \(y\). Remember, multiplying both sides of an inequality by a negative number reverses the inequality sign: \(-2 \cdot -\frac{1}{2} y \geq -2 \cdot 1\longrightarrow y \geq -2\).
03

- Write the Inequality Solution

The inequality solution is \(y \geq -2\).
04

- Graph the Inequality on a Number Line

To graph \(y \geq -2\), draw a number line. Shade the region to the right of \(-2\) and place a closed circle on \(-2\) to indicate that \(-2\) 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.

Graphing Inequalities
When graphing inequalities, the goal is to visually represent all possible solutions on a number line. First, you need to determine if your inequality represents a greater-than, less-than, greater-than or equal to, or less-than or equal to relationship. This helps in knowing where and how to shade the line.
Inequality Sign Reversal
A crucial concept to understand when solving inequalities is the reversal of the inequality sign. This occurs when both sides of an inequality are multiplied or divided by a negative number. For example, if your inequality is \(-x < 5\) and you divide both sides by \(-1\), the inequality sign flips, resulting in \(x > -5\). Remember this rule to avoid errors in your solutions.
Number Line Representation
A number line is an effective tool for representing inequalities visually. Here are the steps to graph an inequality on a number line:
  • Identify and locate the critical value on the number line.
  • Use a closed circle for inequalities involving \(\textgreater=\) or \(\textless=\), and an open circle for \(\textgreater\) or \(\textless\).
  • Shade the region to the right for greater-than or greater-than or equal to inequalities; shade the left for less-than or less-than or equal to inequalities.
This method makes it easy to see all potential solutions at a glance.
Isolating the Variable
Isolating the variable is usually the first step in solving inequalities. The goal is to get the variable by itself on one side of the inequality.
  • Start by undoing any addition or subtraction around the variable.
  • Next, address any multiplication or division to isolate the variable completely.
For example, in the inequality \(-\frac{1}{2} y + 4 \textless= 5\), you would first subtract 4 from both sides, then multiply both sides by \(-2\), remembering to reverse the inequality sign.
Applying these steps systematically helps in avoiding mistakes and ensuring the variable is properly isolated.

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Most popular questions from this chapter

Determine the number of solutions to the system of equations. $$ \begin{aligned} y &=2^{x+1} \\ -1+\log _{2} y &=x \end{aligned} $$

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A fishing boat travels along the east coast of the United States and encounters the Gulf Stream current. It travels \(44 \mathrm{mi}\) north with the current in \(2 \mathrm{hr}\). It travels \(56 \mathrm{mi}\) south against the current in \(4 \mathrm{hr}\). Find the speed of the current and the speed of the boat in still water.

Use substitution to solve the system for the set of ordered triples \((x, y, \lambda)\) that satisfy the system. $$ \begin{array}{l} 2=2 \lambda x \\ 6=2 \lambda y \\ x^{2}+y^{2}=10 \end{array} $$
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