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Chapter 7: Problem 13

Solve and graph the solution. $$.7 x-9 \leq 5$$

Short Answer

Expert verified
The solution for the inequality is \(x \leq 20\). The graph of this solution includes a closed dot at 20 and a line extending to the left of this point.

Step by step solution

01

Simplifying the Inequality

The first step is to isolate \(x\) on one side of the equation by cancelling out the coefficients and constants from the left side. This is achieved by adding \(9\) to both sides of the inequality. This gives us: \(0.7x \leq 5 + 9\). Simplifying the right side of the equation we get \(0.7x \leq 14\).
02

Solve for x

Now we want to solve for \(x\). To remove the coefficient \(0.7\), we can divide both sides of the equation by \(0.7\), resulting in \(x \leq 14 / 0.7\). After carrying out the division, we find that \(x \leq 20\).
03

Graph the Solution

To depict the solution on a number line, draw a number line and place a closed circle or dot at \(20\), because \(x\) can be equal to \(20\). Draw a line extending to the left of the dot to symbolize that \(x\) can be any number less than or equal to \(20\).

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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 us visually understand the solutions of inequalities. When graphing an inequality like \( x \leq 20 \), you must decide which type of circle or dot to use on the number line. First:
  • Determine whether the inequality is \( < \), \( > \), \( \leq \), or \( \geq \).
  • For \( \leq \) or \( \geq \), use a closed circle or dot. This indicates that the number where the dot or circle is placed is included in the solution set.
  • For \( < \) or \( > \), use an open circle. This denotes that the endpoint is not part of the solution.
Once the correct dot or circle is placed, draw an arrow in the direction of the inequality. For example, with \( x \leq 20 \), draw an arrow extending leftwards indicating all numbers less than 20 are solutions. Graphs like these make it easier to see what numbers satisfy the inequality.
Solving Inequalities
When solving inequalities, your goal is to isolate the variable on one side of the inequality. This is similar to solving regular equations, but with a few differences. The steps are:
  • Start by manipulating the inequality to get the variable on one side. This can involve adding, subtracting, multiplying, or dividing both sides by the same number, just as you would with equations.
  • If you multiply or divide by a negative number, remember to flip the inequality sign to maintain the correct relationship between the two sides.
  • Ensure each step is valid, keeping the inequality balanced just like how an equation must remain balanced.
In the provided example, starting with \( 0.7x - 9 \leq 5 \), we first add 9 to both sides and then divide by 0.7. This results in \( x \leq 20 \), a straightforward inequality. Each operation on the inequality moves toward finding all possible values of \( x \) that satisfy the condition set by the inequality.
Number Line Representation
Translating inequalities onto a number line offers a clear view of possible solutions. A number line is a simple straight line where numbers are displayed in order. To represent an inequality like \( x \leq 20 \):
  • First, draw a horizontal line and mark the numbers on it. Decide an appropriate scale that includes the critical value of interest, here 20.
  • Place a closed circle at 20 if the inequality includes that number, as in \( \leq \). This closed circle indicates that 20 is part of the solution.
  • Next, draw a line or arrow pointing in the direction of the inequality. For \( x \leq 20 \), this is leftward, showing all smaller numbers are included because they satisfy the inequality.
Number line representations are useful because they allow you to directly see all values that are solutions to the inequality, enhancing comprehension.

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Most popular questions from this chapter

Graph the solution set of each system of inequalities, when possible. SEE EXAMPLE 7. (OBJECTIVE 4) $$\left\\{\begin{array}{l} x \geq-1 \\ y>-2 \end{array}\right.$$

Graph each inequality. SEE EXAMPLE 4. (OBJECTIVE 2) $$3 x-4 y>12$$

Graph each inequality. SEE EXAMPLE 4. (OBJECTIVE 2) $$2 x-y \leq 4$$

Solve each system using any method. $$\left\\{\begin{array}{l}\frac{x-3}{2}+\frac{y+5}{3}=\frac{11}{6} \\\\\frac{x+3}{3}-\frac{5}{12}=\frac{y+3}{4}\end{array}\right.$$

Use elimination to solve each system. $$\left\\{\begin{array}{l}x-y=9 \\\\\frac{1}{3} x=\frac{1}{3} y+3\end{array}\right.$$
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