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Chapter 0: Problem 80

Solve each absolute value inequality. $$-3|x+7| \geq-27$$

Short Answer

Expert verified
The solution to the given inequality is \(x\) in the interval \([-16, 2]\)

Step by step solution

01

Distribute the negative

First get rid of the negative multiplier on the left side. It can be done by dividing the whole inequality by -3, but remember that dividing by a negative number switches the direction of the inequality: \[ |x+7| \leq 9 \]
02

Set up two equations

To handle the absolute value, consider the cases when \( x+7 \) is positive or negative. This leads to two separate problems to solve: \( x+7 \leq 9 \) and \( x+7 \geq -9 \)
03

Solve the first equation

Solving \( x+7 \leq 9 \) gives \( x \leq 2 \)
04

Solve the second equation

Solving \( x+7 \geq -9 \) gives \( x \geq -16 \)
05

Interpret the solution

The solution to the original inequality is the intersection of the solutions to these two inequalities, that is \( x \) is in the interval \([-16, 2] \)

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Most popular questions from this chapter

Determine whether each statement makes sense or does not make sense, and explain your reasoning. Because I want to solve \(25 x^{2}-169=0\) fairly quickly, I'll use the quadratic formula.

Use the strategy for solving word problems, modeling the verbal conditions of the problem with a linear inequality. The toll to a bridge is \(\$ 3.00 .\) A three-month pass costs \(\$ 7.50\) and reduces the toll to \(\$ 0.50 .\) A six-month pass costs \(\$ 30\) and permits crossing the bridge for no additional fee. How many crossings per three-month period does it take for the three-month pass to be the best deal?

$$\text { Solve for } C: \quad V=C-\frac{C-S}{L} N$$.

Explain how to multiply rational expressions.

What's wrong with this argument? Suppose \(x\) and \(y\) represent two real numbers, where \(x>y .\) $$\begin{aligned} &2>1\\\ &2(y-x)>1(y-x)\\\ &2 y-2 x>y-x\\\ &\begin{aligned} y-2 x &>-x \\ y &>x \end{aligned} \end{aligned}$$ The final inequality, \(y>x,\) is impossible because we were initially given \(x>y\)
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