Question

1. two inequalities are shown below.

-5 < x + 1 and 2x + 2 < 4
a. write the compound inequality to combine the inequalities above.
b. solve the compound inequality for values of x.
c. graph your solution.

Explanation:

Part A

Step1: Combine inequalities

Since it's an 'and' compound inequality, we write \(-5 < x + 1\) and \(2x+2 < 4\) as \(-5 < x + 1<4 - 2x\) (re - arranged the second inequality \(2x + 2<4\) to \(x<1\) and the first to \(x>-6\)). But the standard way for an 'and' compound inequality with two separate inequalities is \(-5 < x + 1\) and \(2x+2 < 4\) which can also be written as \(-6 < x\) and \(x < 1\) or \(-6 < x<1\)

Part B

Step1: Solve \(-5 < x + 1\)

Subtract \(1\) from all parts of the compound inequality \(-5-1 < x+1 - 1\), so \(x>-6\)

Step2: Solve \(2x + 2<4\)

Subtract \(2\) from both sides: \(2x+2 - 2<4 - 2\), then \(2x<2\). Divide both sides by \(2\): \(x < 1\)

Part C

Step1: Graph on number - line

Draw a number - line. Put an open circle at \(x=-6\) (because \(x>-6\), not equal to \(-6\)) and an open circle at \(x = 1\) (because \(x<1\), not equal to \(1\)). Shade the region between \(-6\) and \(1\)

Answer:

A. \(-6 < x<1\)
B. \(x\) is greater than \(-6\) and less than \(1\)
C. Open circles at \(-6\) and \(1\), shaded region between them on the number - line.