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Chapter 0: Problem 84

Solve each absolute value inequality. $$9 \leq|4 x+7|$$

Short Answer

Expert verified
The solution of the inequality is \(x \leq -4\) or \(x \geq 0.5\).

Step by step solution

01

Understanding Absolute Value Inequality

The absolute value of a number is its distance from zero. When we have an absolute value inequality such as \(9 \leq |4x + 7|\), this means that the value of \(4x + 7\) is either greater or equal to 9 or less or equal to -9.
02

Divide the Inequality into Two Cases

We can divide this inequality into two separate cases: \n 1) \(4x + 7 \geq 9\), \n2) \(4x + 7 \leq -9\). We solve these two separate inequalities to find the solution.
03

Solve the First Inequality

First, subtract 7 from both sides to get \(4x \geq 2\). Then divide by 4 to get \(x \geq 0.5\).
04

Solve the Second Inequality

Subtract 7 from both sides to get \(4x \leq -16\). Then, divide by 4 to get \(x \leq -4\).
05

Write Down the Final Solution

The solution of this inequality is \(x \geq 0.5\) or \(x \leq -4\)

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Most popular questions from this chapter

Exercises \(142-144\) will help you prepare for the material covered in the next section. $$\text { Multiply: }\left(2 x^{3} y^{2}\right)\left(5 x^{4} y^{7}\right)$$

Perform the indicated operations. $$(x-y)^{-1}+(x-y)^{-2}$$

Will help you prepare for the material covered in the first section of the next chapter. If \(y=|x+1|,\) find the value of \(y\) that corresponds to values of \(x\) for each integer starting with -4 and ending with 2

What is a quadratic equation?

In solving \(\sqrt{2 x-1}+2=x,\) why is it a good idea to isolate the radical term? What if we don't do this and simply square each side? Describe what happens.
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