Question7 Solve this equation $\mathrm{a}=2 \quad \mathrm{~b}=3 \quad \mathrm{c}=7$ $|a x-b| \leq c$

Studdy Solution

STEP 1

What is this asking? We're trying to find all possible values of $x$ that satisfy the inequality $|2x - 3| \leq 7$ , given $a = 2$ , $b = 3$ , and $c = 7$ .
It's like a puzzle where we need to find the range of $x$ that keeps the absolute value of $2x - 3$ within a certain distance of zero, specifically less than or equal to 7! Watch out! Remember that absolute value measures the distance from zero, so it can be positive or negative!
Don't forget to consider both cases when solving the inequality.

STEP 2

1. Rewrite the absolute value inequality.
2. Solve the first inequality.
3. Solve the second inequality.
4. Combine the solutions.

STEP 3

Let's rewrite our absolute value inequality as two separate inequalities.
Remember, $|2x - 3| \leq 7$ means that $2x - 3$ is trapped between $-7$ and $7$ , inclusive.
So, we can write this as $-7 \leq 2x - 3 \leq 7$ .

STEP 4

We'll tackle the left side first: $-7 \leq 2x - 3$ .
To isolate $2x$ , we'll add 3 to both sides of the inequality (to add to zero on the right side of the inequality).
This gives us $-7 + 3 \leq 2x - 3 + 3$ , which simplifies to $-4 \leq 2x$ .

STEP 5

Now, we want to isolate $x$ , so we'll divide both sides by 2 (to divide to one).
Since 2 is positive, the inequality sign stays the same.
So, we get $\frac{-4}{2} \leq \frac{2x}{2}$ , which simplifies to $-2 \leq x$ .

STEP 6

Now, let's look at the right side: $2x - 3 \leq 7$ .
We'll add 3 to both sides (to add to zero on the left side of the inequality): $2x - 3 + 3 \leq 7 + 3$ .
This gives us $2x \leq 10$ .

STEP 7

To isolate $x$ , we'll divide both sides by 2 (to divide to one).
Again, since 2 is positive, the inequality sign stays the same.
We get $\frac{2x}{2} \leq \frac{10}{2}$ , which simplifies to $x \leq 5$ .

STEP 8

We found that $-2 \leq x$ and $x \leq 5$ .
This means $x$ is greater than or equal to $-2$ and less than or equal to $5$ .
We can write this combined solution as $-2 \leq x \leq 5$ .
So, $x$ can be any number between $-2$ and $5$ , inclusive!

STEP 9

The solution to the inequality $|2x - 3| \leq 7$ is $-2 \leq x \leq 5$ .

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