91Ó°ÊÓ

Inequalities are like equations, but instead of using an equal sign, they use symbols such as \(<, \leq, \geq, \text{and} >\). Solving inequalities involves similar steps as solving equations: isolating the variable and solving the inequality like an equation. Let's break this down with the provided inequality \(-10 \leq 2+4x\).

First, you need to isolate the variable term \(4x\). This is done by subtracting 2 from both sides of the inequality. This subtraction keeps the inequality balanced while moving closer to solving for \(x\). After simplifying, you get \(-12 \leq 4x\).

Next, divide both sides by 4 to solve for \(x\). Dividing does not change the inequality direction unless you divide by a negative number, which we didn't need to here. This step results in \(-3 \leq x\) or written in standard form for ease of reading, \(x \geq -3\). Now we have the solution!

Graphing the solutions of inequalities is a visual way to show which values of a variable make the inequality true. The solution from our exercise is \(x \geq -3\). This tells us that \(x\) can be \(-3\) or any number greater than \(-3\).

To graph this, we first add

• a solid dot on the number where the inequality starts – the solid dot at \(-3\) indicates that \(-3\) is part of the solution (as \(x \geq -3\) includes \(-3\) itself).

After placing the dot, you would draw a line or arrow extending to the right.

• This line shows all the numbers greater than \(-3\) are solutions.

Using a graph helps you quickly see the range of solutions and can be particularly useful for checking work or conveying information clearly.

When graphing inequalities, the number line acts as a visual guide to represent solutions. It is a straight line with numbers placed at intervals. Each point on the line corresponds to a number.

For our inequality \(x \geq -3\), the number line would start with a solid dot on \(-3\). Solid dots are used for inequalities that include the point (like \(\leq\) or \(\geq\)).

Extending a line to the right from this dot shows that all numbers in that direction are part of the solution.

• If the inequality was just greater than or less than (\(>,<\)), we would instead use an open dot to show the boundary isn't included in the set.

Number lines make it easy to visually perceive the range of possible solutions, especially useful in comparing inequalities or seeing overlaps in compound inequalities.