Question

How do you solve the inequality \[5x + 7 < 2\] and \[5x + 7 > - 18\]?

Answer 1

Hint: An inequality compares two values, showing if one is less than, greater than, or simply not equal to another value. Here we need to solve for ‘x’ which is a variable. Solving the given inequality is very like solving equations and we do most of the same thing but we must pay attention to the direction of inequality\[( \leqslant , > )\]. We have two linear inequalities so we can solve them easily.

Complete step-by-step solution:
Let's take \[5x + 7 < 2\].
We know that the inequality doesn’t change if we add or subtract a number on both side.
We subtract 7 on both sides of the inequality we have,
\[
\Rightarrow 5x + 7 - 7 < 2 - 7 \\
\Rightarrow 5x < - 5 \\
 \]
We divide 5 on both sides we have,
\[
\Rightarrow x < - \dfrac{5}{5} \\
\Rightarrow x < - 1 \\
 \]
Hence the solution of \[5x + 7 < 2\] is \[x < - 1\]. The interval form is \[( - \infty , - 1)\].
Now take \[5x + 7 > - 18\]
Following the same procedure that is subtracting 7 on both sides we have,
 \[
\Rightarrow 5x + 7 - 7 > - 18 - 7 \\
\Rightarrow 5x > - 25 \\
 \]
Dividing 5 on both sides we have,
\[
 \Rightarrow x > \dfrac{{ - 25}}{5} \\
 \Rightarrow x > - 5 \\
 \]
Thus the solution of \[5x + 7 > - 18\] is \[x > - 5\]. The interval form is \[( - 5,\infty )\].

Note: Take a value of ‘x’ in \[( - \infty , - 1)\] and put it in \[5x + 7 < 2\].
Let’s put \[x = - 2\] in \[5x + 7 < 2\].
\[
  5( - 2) + 7 < 2 \\
   - 10 + 7 < 2 \\
   - 3 < 2 \\
 \]
Which is correct. We can also check for second inequality the same way.

We know that \[a \ne b\] says that ‘a’ is not equal to ‘b’. \[a > b\] means that ‘a’ is less than ‘b’. \[a < b\] means that ‘a’ is greater than ‘b’. These two are known as strict inequality. \[a \geqslant b\] means that ‘a’ is less than or equal to ‘b’. \[a \leqslant b\] means that ‘a’ is greater than or equal to ‘b’.
The direction of inequality do not change in these cases:
-Add or subtract a number from both sides.
-Multiply or divide both sides by a positive number.
-Simplify a side.

The direction of the inequality change in these cases:
-Multiply or divide both sides by a negative number.
-Swapping left and right hand sides.