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Chapter 7: Problem 21

Solve the inequality \(|3 x+2| \leq 4\)

Short Answer

Expert verified
Answer: The inequality holds true for \(x\) in the interval \([-2, \frac{2}{3}]\).

Step by step solution

01

Break the absolute value inequality into two separate inequalities

When \((3x+2) \geq 0\), the inequality is \(3x + 2 \leq 4\). When \((3x+2) < 0\), the inequality is \(-(3x + 2) \leq 4\).
02

Solve the first inequality

To solve the inequality \(3x + 2 \leq 4\), we'll first isolate the x term by subtracting 2 from both sides: \(3x \leq 2\). Next, we'll divide both sides by 3: \(x \leq \frac{2}{3}\).
03

Solve the second inequality

To solve the inequality \(-(3x + 2) \leq 4\), we first multiply both sides by -1 and reverse the inequality symbol: \((3x + 2) \geq -4\). Then we isolate the x term by subtracting 2 from both sides: \(3x \geq -6\). Finally, we'll divide both sides by 3: \(x \geq -2\).
04

Combine solutions and write in interval notation

Since \(x \leq \frac{2}{3}\) and \(x \geq -2\), we can write the solution in interval notation as \([-2, \frac{2}{3}]\). This represents the values of \(x\) that satisfy the inequality \(|3x + 2| \leq 4\).

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Most popular questions from this chapter

If \(a\) is proportional to \(b\) state which of the following are true and which are false: (a) \(a\) multiplied by \(b\) is a constant (b) \(a\) divided by \(b\) is a constant (c) \(\sqrt{a}\) is proportional to \(\sqrt{b}\)

Rewrite each of the statements without using a modulus sign: $$ |x| \leq 7.5 $$

If \(2 x^{2}+5 x+2=(x+2) \times\) a polynomial what must be the coefficient of \(x\) in this unknown polynomial?

Verify that the given value is a solution of the given equation. $$ 2 x+3=4, x=\frac{1}{2} $$

Solve the equation \(x^{3}-5 x^{2}+2 x+8=0\)
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