Question

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

Answer 1

`-5 <= x < 2`

`x in [-5, 2]`

Explanation:

Graphing

Let's start with graphing.

You can graph `|2x+3|` first. How do you do this?

The graph of `|x|` looks like this:

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

To graph `|2x + 3|`, we need to adapt the slope of the two "arms" and to shift the "elbow" appropriately.

• 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 `|2x+3|` is:

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

Now, the solution of the equation are all `x` values for which the graph is underneath the line `y = 7`:

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

So, you already see on the graph that the solution is `-5 <= x < 2` or `x in [-5, 2]`.

How to find the same solution without the graph?

Solving

To solve the inequality, you need to evaluate the absolute value `|2x+3|` .

Generally, for an absolute value, you have

`|x| = { (x, x >= 0), (-x, x < 0):}`

In this case, you need to know for which `x` the equation `2x + 3 = 0` has a solution.
We already know that this is the case for `x = - 3 /2`.

So, we have

`|2x+3| = { (2x+3, x >= -3/2), (-(2x+3), x < -3/2):}`

Thus, we need to consider the two cases:

1) `x >= -3/2`:

`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) `x < -3/2`:

`-(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 `x in [-3/2; 2]` or `x in [-5; -3/2)` which can be combined to

`x in [-5; 2]`