Question 55·Easy·Linear Inequalities in One or Two Variables

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Which of the following values of $x$ satisfies the compound inequality

$-4 < 2x + 6 \le 10$ ?

For compound inequalities like $-4 < 2x + 6 \le 10$ , treat them as three-part inequalities and perform the same operation on all three parts step by step. First eliminate constants (like subtracting 6), then isolate $x$ (by dividing), remembering to flip the inequality signs only if you multiply or divide by a negative. Once you have a final interval for $x$ , quickly compare each answer choice to that interval, which is usually faster and less error-prone than plugging each choice directly into the original inequality.

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See the structure of the inequality

Notice that $-4 < 2x + 6 \le 10$ is a three-part inequality. Think of it as saying $2x + 6$ must be greater than $-4$ and at the same time less than or equal to $10$ .

Isolate the expression with $x$

Try to isolate $x$ step by step. What operation will remove the $+6$ from the middle expression $2x + 6$ ? Make sure you apply that operation to all three parts of the inequality.

Finish solving for $x$

After you remove the $+6$ , you will have something like a number $<$ or $\le$ $2x$ $<$ or $\le$ another number. What should you divide by to get $x$ alone, and do you need to flip the inequality signs?

Use the final interval to test choices

Once you have an interval for $x$ (like $a < x \le b$ ), check each answer choice against that interval. Pay close attention to which endpoint is strict ( $<$ ) and which is inclusive ( $\le$ ).

Desmos Guide

Graph the expression as a function

In Desmos, enter the equation y = 2x + 6. This shows the line representing $2x + 6$ for all $x$ .

Add the boundary lines for the inequality

Enter y = -4 and y = 10 as two more equations. These horizontal lines represent the lower and upper bounds for $2x + 6$ in the inequality $-4 < 2x + 6 \le 10$ .

Use a table to test the answer choices

Click on the y = 2x + 6 equation and add a table. In the $x$ -column, type the four answer choices: $-6$ , $-1$ , $4$ , and $7$ . Look at the corresponding $y$ -values in the table and check which row has a $y$ -value that is greater than $-4$ and less than or equal to $10$ . The $x$ in that row is the value that satisfies the compound inequality.

Step-by-step Explanation

Understand the compound inequality

The inequality

-4 < 2x + 6 \le 10

is a compound inequality, meaning $2x + 6$ must be greater than $-4$ and less than or equal to $10$ at the same time. We will solve it as a three-part inequality to find all possible $x$ values.

Subtract 6 from all three parts

To isolate $x$ , first get rid of the $+6$ by subtracting $6$ from every part of the inequality:

-4 - 6 < 2x + 6 - 6 \le 10 - 6

which simplifies to

-10 < 2x \le 4.

The inequality signs stay the same because we are just subtracting a number.

Divide all three parts by 2

Now divide every part by $2$ to solve for $x$ :

\frac{-10}{2} < \frac{2x}{2} \le \frac{4}{2}

which simplifies to

-5 < x \le 2.

We do not flip the inequality signs because we are dividing by a positive number.

Compare the solution interval to the answer choices

The solution set is all real numbers $x$ such that

-5 < x \le 2.

Now check each answer choice:

$-6$ is not greater than $-5$ .
$4$ and $7$ are greater than $2$ .
$-1$ is greater than $-5$ and less than or equal to $2$ .

So the only value that satisfies the compound inequality is $-1$ (choice B).

Question 55·Easy·Linear Inequalities in One or Two Variables

Strategy

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Desmos Guide

Step-by-step Explanation

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Strategy

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Desmos Guide

Step-by-step Explanation