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Chapter 1: Problem 86

Solve each equation or inequality. $$ |x-2|-3 \leq 4 $$

Short Answer

Expert verified
The solution is \( -5 \leq x \leq 9 \).

Step by step solution

01

- Isolate the absolute value

Add 3 to both sides of the inequality to isolate the absolute value expression: \[ |x-2| - 3 + 3 \ \leq 4 + 3 \]This results in: \[ |x-2| \leq 7 \]
02

- Set up two inequalities

Since \( |x-2| \leq 7 \), we set up two separate inequalities: \[ x-2 \leq 7 \]and\[ x-2 \geq -7 \]
03

- Solve the first inequality

Solve \( x-2 \leq 7 \):Add 2 to both sides: \[ x-2 + 2 \leq 7 + 2 \]This simplifies to: \[ x \leq 9 \]
04

- Solve the second inequality

Solve \( x-2 \geq -7 \):Add 2 to both sides: \[ x-2 + 2 \geq -7 + 2 \]This simplifies to: \[ x \geq -5 \]
05

- Combine the solutions

Combine the solutions from the two inequalities:\[ -5 \leq x \leq 9 \]

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Key Concepts

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

solving inequalities
Solving inequalities might look a bit tricky at first, but with some practice, you'll get the hang of it! The main goal is to find out which values make the inequality true. Here鈥檚 a simple approach:
  • First, rewrite the inequality in a similar form to an equation.
  • Perform the same operation on both sides of the inequality just like you would with equations.
  • Be careful with multiplication or division by negative numbers, as it flips the inequality sign.
Let鈥檚 see this in our example: \[ |x-2|-3 \leq 4 \] To solve, isolate the absolute value expression. Add 3 to both sides: \[ |x-2| - 3 + 3 \leq 4 + 3 \] This gives: \[ |x-2| \leq 7 \] Now, our inequality is ready for the next step.
absolute value
The absolute value of a number or expression shows its distance from zero on the number line, regardless of direction. Absolute value is always non-negative. For example, \[ |3| = 3 \] and \[ |-3| = 3 \].
When dealing with absolute value inequalities, you often break them into two cases to account for both positive and negative distances.
Let鈥檚 explore our exercise further: \[ |x-2| \leq 7 \] For this absolute value inequality:
  • You set up two inequalities: \[ x-2 \leq 7 \] and \[ x-2 \geq -7 \]
  • These cover all the possible cases of absolute value for any real number.
compound inequalities
A compound inequality combines two inequalities into one statement by using the words 'and' or 'or'. In our exercise, we use 'and': \[ x-2 \leq 7 \] and \[ x-2 \geq -7 \]. We solve them separately and then combine the results.

Solve for the first part:
\[ x-2 \leq 7 \] Add 2 to both sides: \[ x \leq 9 \] Solve for the second part:
\[ x-2 \geq -7 \] Add 2 to both sides: \[ x \geq -5 \] Combine these solutions:
\[ -5 \leq x \leq 9 \] This tells us that x can be any number between -5 and 9, inclusive.When solving compound inequalities, it helps to visualize the solution on a number line to confirm your answer.

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Most popular questions from this chapter

Solve each problem. Randall wants to mix 50 lb of nuts worth \(\$ 2\) per \(\mathrm{lb}\) with some nuts worth \(\$ 6\) per \(\mathrm{lb}\) to make a mixture worth \(\$ 5\) per \(1 \mathrm{~b}\). How many pounds of \(\$ 6\) nuts must he use?

Solve each problem. In \(2017,\) two popular brands on Instagram were National Geographic and Nike. National Geographic had 8.0 million more followers than Nike. Together, the two brands had 158.8 million followers. How many followers did each brand have?

Determine the number of ounces a filled carton of the given size may contain for the given relative error. $$ \left|\frac{x-x_{t}}{x_{t}}\right|=\text { relative error in } x $$ 32-oz carton; relative error no greater than 0.02

Solve each equation or inequality. $$ |6 x-2|=0 $$

Solve each equation or inequality. $$ |10 x+7|+3<1 $$
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