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Chapter 2: Problem 42

For exercises 37-52, (a) solve. (b) use a number line graph to represent the solution. (c) check the direction of the inequality sign. $$ -x+7 \leq-15 $$

Short Answer

Expert verified
x \geq 22.

Step by step solution

01

- Isolate the Variable

Start by isolating the variable term on one side of the inequality. Given inequality is \( -x+7 \leq-15 \). Subtract 7 from both sides:\[ -x+7-7 \leq -15-7 \]Simplify:\[ -x \leq -22 \]
02

- Solve for the Variable

To solve for \( x \), we need to divide both sides by -1. Remember, dividing by a negative number flips the inequality sign:\[ x \geq 22 \]
03

- Graph the Solution on a Number Line

Draw a number line. Mark the point 22 with a closed circle (since the inequality includes equality). Shade the number line to the right of 22 to indicate \( x \geq 22 \).
04

- Check the Direction of the Inequality Sign

To verify, check a number greater than 22 in the original inequality. For example, if \( x = 23 \), then \( -23 + 7 \leq -15 \)\[ -16 \leq -15 \]This is true, so the direction of the inequality sign is correct.

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Key Concepts

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

number line
A number line is a visual representation of numbers in order. It helps to easily see the sequence and relationship between numbers.
To draw a number line for inequalities, start by sketching a horizontal line. Place important numbers, like the solutions of an inequality, on this line. In our exercise, the key number is 22.
Use a closed circle to mark this number on the line, since our inequality includes 22. Then, shade to the right of 22 to show all numbers greater than or equal to it. This shaded part represents the set of all possible solutions for the variable.
inequality sign
An inequality sign shows the relationship between two expressions. It tells us whether one side is greater than, less than, or equal to the other. Common inequality signs include:
  • <: less than
  • \leq: less than or equal to
  • >: greater than
  • \geq: greater than or equal to
In our exercise, we start with \(-x + 7 \leq -15\). After isolating and solving for x by dividing by a negative number, our inequality sign flips. The new inequality sign is \(\geq \).
variable isolation
Variable isolation means getting the variable alone on one side of the inequality. This helps us find the solution.
In our exercise, we start with \(-x + 7 \leq -15\). Follow these steps to isolate x:
1. Subtract 7 from both sides:\(-x + 7 - 7 \leq -15 - 7\)
This simplifies to \(-x \leq -22\). 2. Next, divide both sides by -1. Remember, dividing by a negative flips the inequality sign:\( x \geq 22 \).
Now the variable x is isolated, and we have our solution: \(\geq 22\).
graphing inequalities
Graphing inequalities on a number line visually shows the range of solutions.
Here are the steps to graph our solution \( x \geq 22 \):
  • Draw a horizontal number line.
  • Mark 22 on the line with a closed circle, indicating that 22 is included in the solution.
  • Shade the line to the right of 22. This shading shows that all numbers greater than or equal to 22 satisfy the inequality.
Checking the direction of the shading is important. By using a test point from the shaded area, we can double-check if it indeed satisfies the original inequality. This confirms that the solution is correct.

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Most popular questions from this chapter

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For exercises 37-52, (a) solve. (b) use a number line graph to represent the solution. (c) check the direction of the inequality sign. $$ 4(2 x-6) \leq 5(x-9) $$
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