Linear Equations and Inequalities - Quantitative Reasoning

1. Identify the solution set for $|x-2|\le 5$.

   $-3\le x\le 7$. Rewriting $|x-2| \le 5$ as $-5 \le x-2 \le 5$ and adding 2 to all parts solves the inequality.

2. Identify the $y$-intercept of the line $3x+2y=8$.

   $4$. Rewriting $3x + 2y = 8$ in slope-intercept form reveals the y-intercept as 4.

3. Identify the equation of the line with slope $2$ and $y$-intercept $-3$.

   $y=2x-3$. Substituting slope 2 and y-intercept $-3$ into the slope-intercept form gives the equation.

4. What happens to an inequality when you multiply or divide both sides by a negative number?

   Reverse the inequality sign. Multiplying or dividing an inequality by a negative number reverses the direction to preserve the inequality's validity.

5. What is the condition for two nonvertical lines to be parallel in terms of slopes $m_1$ and $m_2$?

   $m_1=m_2$. Nonvertical parallel lines share the same slope, ensuring they maintain a constant distance without intersecting.

6. What is the slope of the line given by $Ax+By=C$ (with $B\ne 0$)?

   $m=-\frac{A}{B}$. Rearranging the standard form to slope-intercept yields the slope as the negative ratio of coefficients $A$ to $B$.

7. Identify the solution for $\frac{x}{4}+3=5$.

   $x=8$. Subtracting 3 from both sides and multiplying by 4 solves for $x$.

8. What is the equation of a horizontal line passing through $y=b$?

   $y=b$. A horizontal line includes all points with the constant y-coordinate $b$, yielding a slope of zero.

9. What inequality is equivalent to $|x|<a$ for $a>0$?

   $-a<x<a$. The absolute value inequality $|x| < a$ means $x$ lies strictly between $-a$ and $a$.

10. What is the slope-intercept form of a line?

    $y=mx+b$. This form directly incorporates the slope $m$ and y-intercept $b$ to define the linear equation.

11. What inequality is equivalent to $|x|\le a$ for $a\ge 0$?

    $-a\le x\le a$. The absolute value inequality $|x| \le a$ includes all $x$ from $-a$ to $a$ inclusive.

12. What is the point-slope form of a line through $(x_1,y_1)$ with slope $m$?

    $y-y_1=m(x-x_1)$. This form utilizes a known point $(x_1, y_1)$ and the slope $m$ to express the line's equation.

13. Identify the equation of the line through $(1,4)$ with slope $-2$.

    $y=-2x+6$. Using point-slope form with point $(1,4)$ and slope $-2$, then simplifying, yields the equation.

14. Identify the slope of the line through $(2,5)$ and $(6,1)$.

    $m=-1$. Applying the slope formula to the points $(2,5)$ and $(6,1)$ gives the value.

15. What compound inequality is equivalent to $|x|>a$ for $a>0$?

    $x<-a$ or $x>a$. The absolute value inequality $|x| > a$ indicates $x$ is either less than $-a$ or greater than $a$.

16. What compound inequality is equivalent to $|x|\ge a$ for $a\ge 0$?

    $x\le -a$ or $x\ge a$. The absolute value inequality $|x| \ge a$ includes $x$ at or below $-a$ or at or above $a$.

17. Identify the solution set for $5x+1\le 16$.

    $x\le 3$. Subtracting 1 and dividing by 5, with the inequality direction unchanged, yields the solution set.

18. What is the standard form of a linear equation in two variables?

    $Ax+By=C$. This general form represents linear equations where $A$, $B$, and $C$ are constants with $A$ and $B$ not both zero.

19. Identify the solution set for $-2x>6$.

    $x<-3$. Dividing by $-2$ reverses the inequality, resulting in the solution set.

20. What is the equation of a vertical line passing through $x=a$?

    $x=a$. A vertical line consists of all points sharing the fixed x-coordinate $a$, resulting in an undefined slope.

21. Identify the solution set for $3<2x+1\le 9$.

    $1<x\le 4$. Subtracting 1 from all parts and dividing by 2 solves the compound inequality.

22. Identify the solution for $3x-5=16$.

    $x=7$. Adding 5 to both sides and dividing by 3 isolates $x$ to find the solution.

23. What is the slope between points $(x_1,y_1)$ and $(x_2,y_2)$ with $x_2\ne x_1$?

    $m=\frac{y_2-y_1}{x_2-x_1}$. The slope formula computes the ratio of the change in $y$-coordinates to the change in $x$-coordinates between two points.

24. Identify the solution for $2(x-3)=10$.

    $x=8$. Dividing both sides by 2 and adding 3 isolates $x$.

25. What is the condition for two nonvertical lines to be perpendicular in terms of slopes $m_1$ and $m_2$?

    $m_1m_2=-1$. Nonvertical perpendicular lines have slopes that are negative reciprocals, with their product equaling $-1$.

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