Question

Solve the inequality. Write a sentence that describes the solution. $$ -13 \leq 5-2 x<9 $$

Short answer

The solution to the inequality \(-13 \leq 5-2 x<9\) is \(x \geq 9\).

Step-by-step solution

Analyze the inequality

The inequality is \(-13 \leq 5-2 x<9\). This can be split into two separate inequalities: \(-13 \leq 5-2 x\) and \(5-2 x<9\).

Solve first inequality

Start with the inequality \(-13 \leq 5-2 x\). In order to isolate the variable, subtract 5 from both sides, which gives: \(-13-5 \leq -2x\). Simplify this to \(-18 \leq -2x\). To further simplify, divide both sides by -2. Remember to switch the sign when dividing or multiplying both sides of inequality by a negative number. This gives \(x \geq 9\).

Solve second inequality

Now consider the inequality \(5-2x<9\). Subtract 5 from both sides to get \(-2x<4\). Divide both sides by -2 and switch the sign to result in \(x> -2\).

Combine the Solutions

The solution for \(x\) must satisfy both \(x \geq 9\) and \(x > -2\). Since \(9 > -2\) the only possible solution is \(x \geq 9\), no values which are \(x> -2\) would satisfy \(x \geq 9\) unless they were also greater than or equal to 9.

Key concepts

Compound Inequalities

Understanding compound inequalities is key to solving problems that have multiple relationships between variables. A compound inequality consists of two distinct inequalities joined by the word 'and' or the word 'or'.

When the inequalities are connected by 'and', the solution must satisfy both inequalities simultaneously. In the case of our example, the compound inequality \[\begin{equation}-13 \leq 5-2x < 9\end{equation}\]is made up of two inequalities: \[\begin{equation}-13 \leq 5-2x\end{equation}\]and \[\begin{equation}5-2x < 9\end{equation}\]
.The solutions to each part of the compound inequality must overlap, meaning we are looking for values of 'x' that make both parts true. After solving each part, we combine the solutions to find the set of all 'x' that fulfills the entire compound inequality.

Inequality Notation

Inequality notation is a shorthand way of representing the range of values that a variable can take. It's important to be comfortable with inequality symbols, such as '<' (less than), '<=' (less than or equal to), '>' (greater than), '>=' (greater than or equal to).

In the given exercise, we are dealing with a 'less than or equal to' (\[\begin{equation}\leq\end{equation}\]) on one end and a 'less than' (\[\begin{equation}<\end{equation}\]) on the other, denoting that the variable 'x' should be greater than or equal to a number on one side and less than another number on the other.
.It's also crucial to remember that the inequality direction changes when both sides are multiplied or divided by a negative number. This fundamental rule helps prevent errors when isolating the variable in each part of the compound inequality.

Isolating Variables

Isolating the variable is a process used to find the value(s) of a variable that make an equation or inequality true. This generally involves performing the same operation on both sides of the equation or inequality to get the variable on one side by itself.

For instance, the initial step in solving each inequality of a compound inequality is to isolate 'x'.

For the First Inequality:

Start with \[\begin{equation}-13 \leq 5-2x\end{equation}\]You isolate 'x' by subtracting '5' from both sides and then dividing by '-2', bearing in mind to reverse the inequality sign:\[\begin{equation}x \geq 9\end{equation}\]
.

For the Second Inequality:

From \[\begin{equation}5-2x < 9\end{equation}\]You perform similar steps, subtract '5' and divide by '-2', to isolate 'x', resulting in:\[\begin{equation}x > -2\end{equation}\]
.Once 'x' is isolated in each inequality, you can combine the solutions to find the final range of values that 'x' can take.