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Chapter 6: Problem 64

Solve each inequality and graph the solution set on a number line. \(3 \leq 4 x-3<19\)

Short Answer

Expert verified
The solution to the compound inequality \(3 \leq 4x - 3 < 19\) is \(1.5 \leq x < 5.5\).

Step by step solution

01

Solve the First Inequality

Start with the first part of the inequality, \(3 \leq 4x-3\). To solve this inequality, first isolate the term with \(x\) by adding \(3\) on both sides. This gives us \(6 \leq 4x\). Then dividing by \(4\) on both sides, we get \(x \geq 1.5\).
02

Solve the Second Inequality

Now move on to the second part of the inequality, \(4x-3<19\). Here too, isolate the term with \(x\) by adding \(3\) on both sides to give \(4x < 22\). Divide both sides by \(4\) to obtain \(x < 5.5\)
03

Interpret the Combined Inequality

The compound inequality is true if both \(x \geq 1.5\) and \(x < 5.5\). Therefore, the solution set consists of all real numbers \(x\) where \(1.5 \leq x < 5.5\)
04

Graph the Solution

Drawing a number line from 1 to 6, the solution to this compound inequality can be represented graphically on a number line, where \(x\) is greater than or equal to 1.5 and less than 5.5. A closed circle on 1.5 denotes that the value of 1.5 is included in the solution set, while an open circle on 5.5 means that the value of 5.5 is not included in the solution set

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