Linear Equations and Inequalities - ISEE Upper Level Mathematics Achievement

1. What is the solution set meaning of $x\le 3$?

   All real numbers less than or equal to $3$. This inequality includes all values of $x$ that are $3$ or smaller on the real number line.

2. What is $x$ if $3x-7=11$?

   $x=6$. Add 7 to both sides, then divide by 3 to isolate $x$.

3. What does it mean if simplifying an equation gives $0=5$?

   No solution (inconsistent equation). Simplifying to $0=5$ indicates a contradiction, meaning no value satisfies the equation.

4. What is the point-slope form of a linear equation?

   $y-y_1=m(x-x_1)$. This form defines a line using slope $m$ and a point $(x_1, y_1)$ on the line.

5. What is $x$ if $\frac{x}{4}-3=2$?

   $x=20$. Add 3 to both sides, then multiply by 4 to clear the fraction and isolate $x$.

6. What is the solution to the inequality $-3x\ge 12$?

   $x\le -4$. Divide both sides by -3, reversing the inequality sign due to the negative divisor.

7. What is the solution to the inequality $2x-5<9$?

   $x<7$. Add 5 to both sides, then divide by 2, keeping the inequality direction the same.

8. What is the rule for solving $ax=b$ when $a\ne 0$?

   $x=\frac{b}{a}$. Dividing both sides by $a$ isolates $x$, yielding the solution directly.

9. What does it mean if simplifying an equation gives $0=0$?

   Infinitely many solutions (all real numbers). Simplifying to $0=0$ indicates an identity equation true for every real number.

10. What is the solution to the compound inequality $-2\le x+1<5$?

    $-3\le x<4$. Subtract 1 from all parts to isolate $x$, maintaining the compound inequality structure.

11. What is the solution to the inequality $\frac{x-1}{2}>3$?

    $x>7$. Multiply both sides by 2, then add 1, keeping the inequality direction unchanged.

12. What is the rule for distributing $a(b+c)$?

    $a(b+c)=ab+ac$. The distributive property multiplies the factor outside the parentheses by each term inside.

13. What is the rule for combining like terms such as $3x-5x$?

    Add coefficients: $3x-5x=(3-5)x=-2x$. Like terms with the same variable can be combined by adding or subtracting their coefficients while keeping the variable.

14. What is the rule for clearing denominators in $\frac{x}{3}+2=5$?

    Multiply every term by the LCD (here, multiply both sides by $3$). Multiplying through by the least common denominator eliminates fractions, simplifying the equation.

15. What is the solution set meaning of $x>3$?

    All real numbers greater than $3$. This inequality represents all values of $x$ that exceed $3$ on the real number line.

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

    The inequality sign reverses direction. Multiplying or dividing by a negative number reverses the inequality to maintain the correct relationship between values.

17. What is the rule for distributing a negative sign: $-(b-c)$?

    $-(b-c)=-b+c$. Distributing the negative sign multiplies each term inside the parentheses by $-1$, changing the signs accordingly.

18. What is the slope-intercept form of a linear equation?

    $y=mx+b$. This form expresses a line with slope $m$ and y-intercept $b$.

19. What is $x$ if $2x+3=5x-9$?

    $x=4$. Subtract $2x$ from both sides, add 9 to both sides, then divide by 3 to find $x$.

20. What is $x$ if $3(x+4)-2x=10$?

    $x=-2$. Distribute the 3, combine like terms, then subtract 12 from both sides to isolate $x$.

21. What is $x$ if $5(x-2)=15$?

    $x=5$. Distribute the 5, add 10 to both sides, then divide by 5 to solve for $x$.

22. What is the slope formula between $(x_1,y_1)$ and $(x_2,y_2)$?

    $m=\frac{y_2-y_1}{x_2-x_1}$. The slope is calculated as the change in y divided by the change in x between two points.

23. What is $x$ if $7-2(x+1)=1$?

    $x=2$. Distribute the -2, combine like terms, subtract 5 from both sides, then divide by -2 to solve.

24. What operation must you do to both sides of an equation to keep it equivalent?

    Add, subtract, multiply, or divide both sides by the same nonzero number. Performing the same arithmetic operation on both sides preserves the equality and maintains equivalent equations.

25. What is the meaning of a solution to an equation in one variable?

    A value that makes the equation true when substituted. A solution satisfies the equation by making both sides equal upon substitution.

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