Linear Equations and Inequalities - ISEE Upper Level: Mathematics Achievement

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

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

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

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

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.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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.

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

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

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.

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