# How do you graph and solve #|2x+3| <= 7#?

##### 1 Answer

#### Explanation:

**Graphing**

Let's start with graphing.

You can graph

The graph of

graph{|x| [-15, 15, -3, 12]}

To graph

- the slope is being described by the factor in front of
#x# . Here, the slope is#2# . - to find the "elbow", set
#2x + 3 = 0# and solve for#x# . Thus, the "elbow" is at#(-3/2, 0)#

Thus, the graph of

graph{|2x+3| [-15, 15, -3, 12]}

Now, the solution of the equation are all

graph{(y - |2x+3|)(y-7) = 0 [-15, 15, -3, 12]}

So, you already see on the graph that the solution is

How to find the same solution without the graph?

**Solving**

To solve the inequality, you need to evaluate the absolute value

Generally, for an absolute value, you have

In this case, you need to know for which

We already know that this is the case for

So, we have

Thus, we need to consider the two cases:

1)

#2x + 3 <= 7#

#<=> x <= 2 # Don't forget to take a look at the condition

#x >= -3/2# which needs to hold at the same time.

Here, the solution is#-3/2 <= x <= 2# or#x in [-3/2; 2]#

2)

#-(2x + 3) <= 7#

#<=> -2x - 3 <= 7 #

#<=> -2x <= 10 # ... divide by

#-2# and don't forget to flip the inequality sign (this needs to be done if multiplying with or dividing by a negative number)...

#<=> x >= -5# Don't forget to take a look at the condition

#x < -3/2# which needs to hold at the same time.

Thus, the solution for this case is#-5 <= x < -3/2# or#x in [-5; -3/2)#

In total, we have